# Ch07. Integrals

## Achievements

The following badges can be earned while learning.

## Certificate

Certificate on successful completion of this course.

In this online course, you will learn integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them

{\displaystyle \int{\frac{dx}{x^2 \pm a^2}} } , {\displaystyle \int{\frac{dx}{\sqrt{x^2 \pm a^2}}} } , {\displaystyle \int{\frac{dx}{\sqrt{a^2 - x^2}}} } , {\displaystyle \int{\frac{dx}{ax^2+bx+c}} } , {\displaystyle \int{\frac{dx}{\sqrt{ax^2+bx+c}}} } , {\displaystyle \int{\frac{px+q}{ax^2+bx+c}}dx } , {\displaystyle \int{\frac{px+q}{\sqrt{ax^2+bx+c}}}dx } , {\displaystyle \int{\sqrt{a^2 \pm x^2}}dx } , {\displaystyle \int{\sqrt{x^2-a^2}}dx } , {\displaystyle \int{\sqrt{ax^2+bx+c}}dx } , {\displaystyle \int{(px+q) \sqrt{ax^2+bx+c}}dx }

For further understanding of concepts and for examination preparation, practice questions based on the above topics are discussed in the form of assignments that has questions from NCERT Textbook exercise, NCERT Examples, Board’s Question Bank, RD Sharma, NCERT Exemplar etc. instead of only one book. The PDF of assignments can be downloaded within the course.
“Limit of Sum” and “Properties of Integrals” are not included in this course.

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The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### ASSIGNMENT – 1

1.     {\displaystyle \int{x^2\left(1-\frac{1}{x^2}\right)dx}=\frac{x^3}{3}-x+C } (NCERT Exercise 7.1 Q7)
2.     {\displaystyle {\int\left(\sqrt x-\frac{1}{\sqrt x}\right)}^2dx=\frac{x^2}{2}+log{|}x|-2x+C } (NCERT Exercise 7.1 Q10)
3.     {\displaystyle \int\left(8^x+x^8+\frac{8}{x}+\frac{x}{8}\right)dx=\frac{8^x}{log{8}}+\frac{x^9}{9}+8log{|}x|+\frac{x^2}{16}+C }
4.     {\displaystyle \int{(e^{alog{x}}+e^{xlog{a}})dx}=\frac{x^{a+1}}{a+1}+\frac{a^x}{log{a}}+C }
5.     {\displaystyle \int\left(\frac{cos{2}x+2{sin}^2{x}}{{cos}^2{x}}\right)dx=tan{x}+C }
6.     {\displaystyle \int{(x^c+c^x)dx}=\frac{x^{c+1}}{c+1}+\frac{c^x}{log{c}}+C }
7.     {\displaystyle \int\frac{x^2+3x+4}{\sqrt x}dx=\frac{2}{5}x^\frac{5}{2}+2x^\frac{3}{2}+8\sqrt x+C } (NCERT Exercise 7.1 Q12)
8.     {\displaystyle \int\left(\frac{2a}{\sqrt x}-\frac{b}{x^2}+3c\sqrt[3]{x^2}\right)dx=4a\sqrt x+\frac{b}{x}+\frac{9cx^\frac{5}{3}}{5}+C }
9.     {\displaystyle \int{\frac{x^3-x^2+x-1}{x-1}dx}=\frac{x^3}{3}+x+C } (NCERT Exercise 7.1 Q13)
10.     {\displaystyle \int\frac{{sec}^2{x}}{\mathrm{cose}\mathrm{c}^2x}dx=tan{x}-x+C } (NCERT Exercise 7.1 Q19)
11.     {\displaystyle \int\sqrt{1+sin{2}x}dx=-cos{x}+sin{x}+C }
12.     {\displaystyle \int{sin{2}x.dx}=-\frac{1}{2}cos{2}x+C } (NCERT Exercise 7.1 Q1)
13.     {\displaystyle \int{\frac{1}{\sqrt x+\sqrt{x-1}}dx}=\frac{2}{3}x^\frac{3}{2}-\frac{2}{3}(x-1)^\frac{3}{2}+C }
14.     {\displaystyle {\int\left(x-\frac{1}{2}\right)}^3dx=\frac{1}{4}\left(x-\frac{1}{2}\right)^4+C }
15.     {\displaystyle \int\frac{e^x}{a^x}dx=\frac{\left(\frac{e}{a}\right)^x}{log(\frac{e}{a)}}+C }
16.     {\displaystyle \int{cos}^2{\alpha}dx=x{cos}^2{\alpha}+C }
17.     {\displaystyle \int\frac{1}{xcos{\alpha}+1}dx=\frac{log{|}xcos{\alpha}+1|}{cos{\alpha}}+C }
18.     {\displaystyle \int{\frac{1}{cos{\alpha}+xsin{\alpha}}dx}=\frac{log{|}cos{\alpha}+xsin{\alpha}|}{sin{\alpha}}+C }
19.     {\displaystyle \int{\frac{sin{x}+cos{x}}{\sqrt{1+sin{2}x}}dx}=x+C }
20.     {\displaystyle \int{({sin}^{-1}}\sqrt x+{cos}^{-1}{\sqrt x})dx=\frac{\pi x}{2}+C }
21.     {\displaystyle \int\frac{1}{1-{sin}^2{x}}dx=tan{x}+C }
22.     {\displaystyle \int\frac{2-3sin{x}}{{cos}^2{x}}dx=2tan{x}-3sec{x}+C } (NCERT Exercise 7.1 Q20)
23. chapter 7 class 12 maths

24.     {\displaystyle \int{e^{-{loge}^x}dx}=-e^{-x}+C }
25.     {\displaystyle \int\sqrt{1+cos{2}x}dx=\sqrt2sin{x}+C }
26.     {\displaystyle \int{{tan}^{-1}{\sqrt{\frac{1-cos{2}x}{1+cos{2}x}}}dx}=\frac{x^2}{2}+C }
27.     {\displaystyle \int{{tan}^{-1}(sec{x}+tan{x})}dx=\frac{\pi}{4}x+\frac{x^2}{4}+C }
28.     {\displaystyle \int{{tan}^{-1}{\sqrt{\frac{1-sin{x}}{1+sin{x}}}}dx}=\frac{\pi}{4}x-\frac{x^2}{4}+C }
29.     {\displaystyle {\int\left(\sqrt{ax}-\frac{1}{\sqrt{ax}}\right)}^2dx=\frac{ax^2}{2}+\frac{log{|}x|}{a}-2x+C }
30.     {\displaystyle \int{\frac{sin{4}x}{sin{2}x}dx}      =     sin{2}x+C }
31.     {\displaystyle \int{e^{2x+3}dx}=\frac{1}{2}e^{2x+3}+C } (NCERT Exercise 7.2 Q16)
32.     {\displaystyle \int{{sec}^2(7-4x)dx=-\frac{1}{4}tan(7-4x)+C } } (NCERT Exercise 7.2 Q22)
33.     {\displaystyle \int\frac{cos{2}x-cos{2}\alpha}{cos{x}-cos{\alpha}}dx=2(sin{x}+xcos{\alpha})+C }
34.     {\displaystyle \int{{sin}^{-1}(cos{x})dx}=\frac{\pi x}{2}-\frac{x^2}{2}+C }
35.     {\displaystyle \int{{sin}^2(2x+5)dx=\frac{x}{2}-\frac{1}{8}sin(4x+10)+C } }
36.     {\displaystyle \int{{sin}^3(2x+1)dx}=-\frac{1}{2}cos(2x+1)+\frac{1}{6}{cos}^3(2x+1)+C }
37.     {\displaystyle \int{sin}^4xdx=\frac{3x}{8}-\frac{1}{4}sin{2}x+\frac{1}{32}sin{4}x+C }
38.     {\displaystyle \int{sin{x}.sin{2}xdx}       =       -\frac{1}{2}\left(\frac{sin{3}x}{3}-sin{x}\right)+C }
39.     {\displaystyle \int{sin{3}xcos{4}x}dx=-\frac{1}{14}cos{7}x+\frac{1}{2}cos{x}+C }
40.     {\displaystyle \int{{tan}^2(2x-3)dx=\frac{1}{2}tan(2x-3)-x+C } } (NCERT Exercise 7.2 Q21)
41.     {\displaystyle \int{{sin}^2{x}{cos}^4{x}}dx=\frac{1}{32}\left[2x+\frac{1}{2}sin{2}x-\frac{1}{2}sin{4}x-\frac{1}{6}sin{6}x\right]+C }
42.     {\displaystyle \int{cos{2}xcos{4}xcos{6}x}dx=\frac{1}{4}\left[\frac{1}{12}sin{1}2x+x+\frac{1}{8}sin{8}x+\frac{1}{4}sin{4}x\right]+C }
43.     {\displaystyle \int{sin{x}sin{2}xsin{3}xdx}=\frac{1}{4}\left[\frac{1}{6}cos{6}x-\frac{1}{4}cos{4}x-\frac{1}{2}cos{2}x\right]+C }
44.     {\displaystyle \int{{tan}^{-1}{\sqrt{\frac{1-sin{x}}{1+sin{x}}}}dx}     =   \frac{\pi}{4}x-\frac{x^2}{4}+C }
45.     {\displaystyle \int{\frac{e^{5log{x}}-e^{4log{x}}}{e^{3log{x}}-e^{2log{x}}}dx}=\frac{x^3}{3}+C }
46.     {\displaystyle \int\frac{dx}{\sqrt{1-2x}+\sqrt{3-2x}}=\frac{1}{6}(1-2x)^\frac{3}{2}-\frac{1}{6}(3-2x)^\frac{3}{2}+C }
47.     {\displaystyle \int{{tan}^{-1}(cot{x})dx}       =       \frac{\pi}{2}x-\frac{x^2}{2}+C }
48.     {\displaystyle \int_{0}^{\pi}({sin}^2{\frac{x}{2}}-{cos}^2{\frac{x}{2}})dx=0 }
49.     {\displaystyle \int_{0}^{1}\frac{dx}{\sqrt{1+x}-\sqrt x}=\frac{4\sqrt2}{3} }
50.     {\displaystyle \int\frac{1}{1+cos{x}}dx=tan{\frac{x}{2}}+C }
51.     {\displaystyle \int{\frac{1}{1+sin{x}}dx}=tan{x}-sec{x}+C }
52.     {\displaystyle \int{\frac{sin{x}}{1+sin{x}}dx}=sec{x}-tan{x}+x+C }
53.     {\displaystyle \int\frac{cos{x}}{1+cos{x}}dx= x-tan{\frac{x}{2}}+C }
54.     {\displaystyle \int\frac{sin(x-\alpha)}{sin(x+\alpha)}dx=xcos{2}\alpha-sin{2}\alpha l o g{|}sin(x+\alpha)|+C }
55.     {\displaystyle \int{\frac{cos(x+a)}{cos(x-a)}dx}=xcos{2}a-sin{2}alog{|}sec(x-a)|+C }
56.     {\displaystyle \int{\frac{sin{x}}{sin(x+a)}dx= x c o s{a}-sin{a}.log{|}sin(x+a)|+C } } (NCERT Example Q6(ii))
57.     {\displaystyle \int{\frac{1}{sin(x-a)sin(x-b)}dx}=\frac{1}{sin(a-b)}log{\left|\frac{sin(x-a)}{sin(x-b)}\right|}+C }
58.     {\displaystyle \int{\frac{sin{x}}{sin(x-a)}dx= s i n{a}log{|}sin(x-a)|+xcos{a}+C } }
59.     {\displaystyle \int\frac{1}{cos(x+a)cos(x+b)}dx=\frac{1}{sin(a-b)}log{\left|\frac{cos(x+b)}{cos(x+a)}\right|}+C }
60.     {\displaystyle \int{\frac{sin(x+a)}{sin(x+b)}dx}= xcos(a-b)+sin(a-b)log{|}sin(x+b)|+C }
61.     {\displaystyle \int{\frac{cos(x+a)}{sin(x+b)}dx}= c o s(a-b)log{|}sin(x+b)|-sin(a-b).x+C }

### ASSIGNMENT – 2

1.     {\displaystyle \int{2x.sin(x^2+1)dx}=-cos(x^2+1)+C }
2.     {\displaystyle \int\frac{{tan}^4{\sqrt x}{sec}^2{\sqrt x}}{\sqrt x}dx=\frac{2}{5}{tan}^5{\sqrt x}+C }
3.     {\displaystyle \int\frac{sin({tan}^{-1}{x})}{1+x^2}dx=-cos({tan}^{-1}{x})+C }
4.     {\displaystyle \int\frac{(log{x})^2}{x}dx=\frac{1}{3}(log{|}x|)^3+C }
5.     {\displaystyle \int{sin}^3{x}{cos}^2{x}dx=-\frac{1}{3}{cos}^3{x}+\frac{1}{5}{cos}^5{x}+C }
6.     {\displaystyle \int{{cos}^5{x}dx}=sin{x}-\frac{2}{3}{sin}^3{x}+\frac{1}{5}{sin}^5{x}+C }
7.     {\displaystyle \int{cot{x}logsin{x}dx}=\frac{1}{2}(logsin{x})^2+C }
8.     {\displaystyle \int{\frac{sin{x}}{1+cos{x}}dx}=-log{|}1+cos{x}|+C }
9.     {\displaystyle \int{\frac{1}{xlog{x}log(log{x})}dx}     =    log{|}log(log{x})|+C }
10.     {\displaystyle \int{\frac{e^{\sqrt x}}{\sqrt x}dx}=2e^{\sqrt x}+C }
11.     {\displaystyle \int{sin}^3{x}dx=-\frac{3}{4}cos{x}+\frac{1}{12}cos{3}x+C } or {\displaystyle -cos{x}+\frac{1}{3}{cos}^3{x}+C }
12.     {\displaystyle \int\frac{1}{x(2+3log{x})}dx     =    \frac{1}{3}log{|}2+3log{x}|+C }
13.     {\displaystyle \int{\frac{1-sin{2}x}{x+{cos}^2{x}}dx}       =   log{|}x+{cos}^2{x}|+C }
14.     {\displaystyle \int{\frac{3ax}{b^2+c^2x^2}dx}       =   \frac{3a}{2c^2}log{|}b^2+c^2x^2|+C }
15.     {\displaystyle \int\frac{sin{2}x}{a^2{sin}^2{x}+b^2{cos}^2{x}}dx    =    \frac{1}{(a^2-b^2)}log{|}a^2{sin}^2{x}+b^2{cos}^2{x}|+C }
16.     {\displaystyle \int{{tan}^8{x}{sec}^4{x}}dx         =   \frac{{tan}^{11}{x}}{11}+\frac{{tan}^9{x}}{9}+C }
17.     {\displaystyle \int{\frac{x^2}{1+x^3}dx}    =    \frac{1}{3}log{|}1+x^3|+C }
18.     {\displaystyle \int\frac{1-cos{x}}{sin{x}}dx=2log{\left|sec{\frac{x}{2}}\right|}+C }
19.     {\displaystyle \int\frac{{sin}^3{x}+{cos}^3{x}}{{sin}^2{x}.{cos}^2{x}}dx=sec{x}-\mathrm{cosec}x+C }
20.     {\displaystyle \int\frac{dx}{sin{x}{cos}^3{x}}=log{|}tan{x}|+\frac{1}{2}{tan}^2{x}+C }
21.     {\displaystyle \int{\frac{1+tan{x}}{1-tan{x}}dx}        =   -log{|}cos{x}-sin{x}|+C }
22.     {\displaystyle \int{\frac{cos{2}x}{(sin{x}+cos{x})^2}dx=log{\left|sin{x}+cos{x}\right|}}+C }
23.     {\displaystyle \int\frac{dx}{x+xlog{x}}=log{|}1+log{x}|+C }
24.     {\displaystyle \int{\frac{1}{1+tan{x}}dx}=\frac{x}{2}+\frac{1}{2}log{|}cos{x}+sin{x}|+C }
25.     {\displaystyle \int\frac{1}{x-\sqrt x}dx=2log{|}\sqrt x-1|+C }
26.     {\displaystyle \int{\frac{1}{{cos}^2{x}(1-tan{x})^2}dx=\frac{1}{1-tan{x}}}+C }
27.     {\displaystyle \int\frac{2cos{x}-3sin{x}}{6cos{x}+4sin{x}}dx=\frac{1}{2}log{|}2sin{x}+3cos{x}|+C }
28.     {\displaystyle \int{\frac{e^{{tan}^{-1}{x}}}{1+x^2}dx}=e^{{tan}^{-1}{x}}+C }
29.     {\displaystyle \int\frac{e^{2x}-1}{e^{2x}+1}dx=log{|}e^x+e^{-x}|+C }
30.     {\displaystyle \int{\sqrt{sin{2}x}cos{2}x}dx=\frac{1}{3}(sin{2}x)^\frac{3}{2}+C }
31.     {\displaystyle \int{\frac{cos{x}}{\sqrt{1+sin{x}}}dx}=2\sqrt{1+sin{x}}+C }
32.     {\displaystyle \int\frac{1}{1+cot{x}}dx=\frac{x}{2}-\frac{1}{2}log{|}cos{x}+sin{x}|+C }
33.     {\displaystyle \int\frac{\sqrt{tan{x}}}{sin{x}cos{x}}dx=2\sqrt{tan{x}}+C }
34.     {\displaystyle \int\frac{(x+1)(x+log{x})^2}{x}dx=\frac{1}{3}(x+log{x})^3+C }
35.     {\displaystyle \int\frac{x^3sin({tan}^{-1}{x^4})}{1+x^8}dx=-\frac{1}{4}cos({tan}^{-1}{x^4})+C }
36.     {\displaystyle \int\frac{10x^9+10^xlog{1}0}{x^{10}+10^x}dx=log{|}10^x+x^{10}|+C }
37.     {\displaystyle \int{sin}^3{x}{cos}^3{x}dx=\frac{1}{6}{cos}^6{x}-\frac{1}{4}{cos}^4{x}+C }
38.     {\displaystyle \int{\frac{1+cot{x}}{x+logsin{x}}dx}      =      log{|}x+logsin{x}|+C }
39.     {\displaystyle \int{{tan}^2{x}{sec}^4{x}dx}         =        \frac{{tan}^5{x}}{5}+\frac{{tan}^3{x}}{3}+C }
40.     {\displaystyle \int e^{log(x+1)-log{x}}dx        =      x+log{x}+C }
41.     {\displaystyle \int{\frac{sin{x}}{sin{2}x}dx}       =       \frac{1}{2}log{|}sec{x}+tan{x}|+C }
42.     {\displaystyle \int{{tan}^4{x}dx}=\frac{1}{3}{tan}^3{x}-tan{x}+x+C }
43.     {\displaystyle \int{\frac{2x+3}{x^2+3x}dx}  =   log{|}x^2+3x|+C }
44.     {\displaystyle \int{\frac{1+cos{x}}{x+sin{x}}dx}    =   log{|}x+sin{x}|+C }
45.     {\displaystyle \int{{cos}^4{2}xdx}=\frac{3x}{8}+\frac{1}{8}sin{4}x+\frac{1}{64}sin{8}x+C }
46.     {\displaystyle \int\frac{{sin}^2{x}}{1+cos{x}}dx=x-sin{x}+C }
47.     {\displaystyle \int\frac{{sin}^6{x}}{{cos}^8{x}}dx       =  \frac{{tan}^7{x}}{7}+C }
48.     {\displaystyle \int\frac{cos{x}-sin{x}}{1+sin{2}x}dx=-\frac{1}{cos{x}+sin{x}}+C }
49.     {\displaystyle \int{{tan}^3{2}xsec{2}x}dx=\frac{1}{6}{sec}^3{2}x-\frac{1}{2}sec{2}x+C }
50.     {\displaystyle \int\frac{dx}{{cos}^2{x}+sin{2}x}        =       \frac{1}{2}log{|}1+2tan{x}|+C }
51.     {\displaystyle \int\frac{cos{2}x}{(cos{x}+sin{x})^2}dx  = \ log{|}cos{x}+sin{x}|+C }
52.     {\displaystyle \int\frac{dx}{sin{x}{cos}^3{x}}  =  \ l\ o\ g{|}tan{x}|+\frac{1}{2}{tan}^2{x}+C }
53.     {\displaystyle \int\frac{x^2+4x}{x^3+6x^2+5}dx  =  \frac{1}{3}log{|}x^3+6x^2+5|+C }
54.     {\displaystyle \int{sec{x}.log(sec{x}+tan{x})dx}\ =\ \frac{(log{|}sec{x}+tan{x}|)^2}{2}+C }
55.     {\displaystyle \int{cot{x}.logsin{x}dx}     =   \frac{(logsin{x})^2}{2}+C }
56.     {\displaystyle \int{{cot}^3{x}{cosec}^4{x}dx}    =  -\left(\frac{{cot}^6{x}}{6}+\frac{{cot}^4{x}}{4}\right)+C }
57.     {\displaystyle \int\frac{x^{e-1}+e^{x-1}}{x^e+e^x}dx        =   \frac{1}{e}log{|}x^e+e^x|+C }
58.     {\displaystyle \int\frac{e^x(1+x)}{{cos}^2(e^xx)}dx   = \ t\ a\ n(xe^x)+C }
59.     {\displaystyle \int{\frac{{cos}^5{x}}{sin{x}}dx} = log{|}sin{x}|+\frac{1}{4}{sin}^4{x}-{sin}^2{x}+C }
60.     {\displaystyle \int\frac{4x+1}{\sqrt{2x^2+x-3}}dx=2\sqrt{2x^2+x-3}+C }
61.     {\displaystyle \int\frac{dx}{e^x+e^{-x}}={tan}^{-1}(e^x)+C }
62.     {\displaystyle \int\cos{6}x\sqrt{1+sin{6}x}dx=\frac{1}{9}(1+sin{6}x)^\frac{3}{2}+C }
63.     {\displaystyle \int{{\cos}^2{x}.e^{logsin{x}}dx=-\frac{1}{4}{cos}^4{x}}+C }
64.     {\displaystyle \int{\frac{\sqrt{tan{x}}}{sin{2}x}dx}=2\sqrt{tan{x}}+C }
65.     {\displaystyle \int\frac{{cos}^9{x}}{sin{x}}dx=log{|}sin{x}|-2{sin}^2{x}+\frac{3}{2}{sin}^4{x}-\frac{2}{3}{sin}^6{x}+\frac{1}{8}{sin}^8{x}+C }
66.     {\displaystyle \int e^{3log{x}}(x^4+1)^{-1}dx=\frac{1}{4}log(x^4+1)+C }
67.     {\displaystyle \int_{0}^{\frac{\pi}{4}}{{sin}^3{2}tcos{2}tdt=\frac{1}{8}} }
68.     {\displaystyle \int_{0}^{\frac{\pi}{2}}\sqrt{sin{\phi}}{cos}^5{\phi}d\phi=\frac{64}{231} }
69.     {\displaystyle \int_{4}^{9}{\frac{\sqrt x}{(30-x^\frac{3}{2})^2}dx}=\frac{19}{99} }
70.     {\displaystyle \int_{0}^{1}\left[xe^x+sin{\frac{\pi x}{4}}\right]dx=1+\frac{4}{\pi}-\frac{2\sqrt2}{\pi} }
71.     {\displaystyle \int_{0}^{\frac{\pi}{2}}{{sin}^3{x}dx=\frac{2}{3}} }
72.     {\displaystyle \int\frac{1}{\sqrt{{sin}^3{x}{cos}^5{x}}}dx=-\frac{2}{\sqrt{tan{x}}}+\frac{2}{3}(tan{x})^\frac{3}{2}+C }
73.     {\displaystyle \int_{0}^{\frac{\pi}{4}}{2{tan}^3{x}dx}=1-log{2} }
74.     {\displaystyle \int\frac{x}{\sqrt{x+4}}dx=\frac{2}{3}\sqrt{x+4}(x-8)+C }
75.     {\displaystyle \int{\frac{2x-1}{2x+3}dx}         =      x-log{|}(2x+3)^2|+C }
76.     {\displaystyle \int{\frac{8x+13}{\sqrt{4x+7}}dx}            =       \frac{1}{3}(4x+7)^\frac{3}{2}-\frac{1}{2}(4x+7)^\frac{1}{2}+C }
77.     {\displaystyle \int{\frac{x}{\sqrt{x+2}}dx}           =          \frac{2}{3}(x+2)^\frac{3}{2}-4(x+1)^\frac{1}{2}+C }
78.     {\displaystyle \int{\frac{x+1}{\sqrt{2x-1}}dx}=\frac{1}{6}(2x-1)^\frac{3}{2}+\frac{3}{2}(2x-1)^\frac{1}{2}+C }
79.     {\displaystyle \int\frac{x^5}{x+1}dx=\frac{x^5}{5}-\frac{x^4}{4}+\frac{x^3}{3}-\frac{x^2}{2}+x-log{|}x+1|+C }
80.     {\displaystyle \int\frac{x^4+1}{x^2+1}dx=\frac{x^3}{3}-x+2{tan}^{-1}{x}+C }
81.     {\displaystyle \int\frac{1}{x^\frac{1}{2}+x^\frac{1}{3}}dx=2\sqrt x-3x^\frac{1}{3}+6x^\frac{1}{6}-6log(1+x^\frac{1}{6})+C }
82.     {\displaystyle \int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}dx=-2\sqrt{1-x}+{cos}^{-1}{\sqrt x}+\sqrt{x-x^2}+C }

### ASSIGNMENT – 3

1.     {\displaystyle \int{xe^x}dx= xe^x-e^x+C }
2.     {\displaystyle \int{xcos{x}dx}= xsin{x}+cos{x}+C }
3.     {\displaystyle \int{xsin{3}x}dx=-\frac{x}{3}cos{3}x+\frac{1}{9}sin{3}x+C }
4.     {\displaystyle \int{xlog{2}x}dx=\frac{x^2}{2}log{2}x-\frac{x^2}{4}+C }
5.     {\displaystyle \int{x^2log{x}dx=}\frac{x^3}{3}log{x}-\frac{x^3}{9}+C }
6.     {\displaystyle \int{x{tan}^{-1}{x}}dx=\frac{x^2}{2}{tan}^{-1}{x}-\frac{x}{2}+\frac{1}{2}{tan}^{-1}{x}+C }
7.     {\displaystyle \int(x^2+1)log{x}dx=\left(\frac{x^3}{3}+x\right)log{x}-\frac{x^3}{9}-x+C }
8.     {\displaystyle \int e^{2x}sin{x}dx=\frac{e^{2x}}{5}(2sin{x}-cos{x})+C }
9.     {\displaystyle \int log{x}dx= xlog{x}-x+C }
10.     {\displaystyle \int{{tan}^{-1}{x}dx=}x{tan}^{-1}{x}-\frac{1}{2}log{(}1+x^2)+C }
11.     {\displaystyle \int_{0}^{1}{{sin}^{-1}{x}dx=\frac{\pi}{2}-1} }
12.     {\displaystyle \int{e^x(sin{x}+cos{x})dx= e^xsin{x}}+C }
13.     {\displaystyle \int\frac{xe^x}{(1+x)^2}dx=\frac{e^x}{1+x}+C }
14.     {\displaystyle \int\frac{(x-4)e^x}{(x-2)^3}dx=\frac{e^x}{(x-2)^2}+C }
15.     {\displaystyle \int\frac{2+sin{2}x}{1+cos{2}x}e^xdx= e^xtan{x}+C }
16.     {\displaystyle \int{e^x\left(\frac{sin{4}x-4}{1-cos{4}x}\right)}dx= e^x.cot{2}x+C }
17.     {\displaystyle \int\frac{(x^2+1)e^x}{(x+1)^2}dx=\left(\frac{x-1}{x+1}\right)e^x+C }
18.     {\displaystyle \int{e^x\left(\frac{1-x}{1+x^2}\right)^2dx=\frac{e^x}{1+x^2}}+C }
19.     {\displaystyle \int_{\frac{\pi}{2}}^{\pi}{e^x\left(\frac{1-sin{x}}{1-cos{x}}\right)}dx= e^\frac{\pi}{2} }

### ASSIGNMENT – 4

1. {\displaystyle \int\frac{dx}{3x^2+13x-10}=\frac{3}{7}log{\left|\frac{6x+6}{6x+20}\right|}+C }
2. {\displaystyle \int\frac{3x-2}{(x+1)^2(x+3)}dx=\frac{11}{4}log{\left|\frac{x+1}{x+3}\right|}+\frac{5}{2(x+1)}+C }
3. {\displaystyle \int\frac{x^2+1}{x^2-5x+6}dx= x-5log{|}x-2|+10log{|}x-3|+C }
4. {\displaystyle \int\frac{x^2+x+1}{(x+2)(x^2+1)}dx=\frac{3}{5}log{|}x+2|+\frac{1}{5}log{|}x^2+1|+\frac{1}{5}{tan}^{-1}{x}+C }
5. {\displaystyle \int\frac{2x-3}{(x^2-1)(2x+3)}dx=\frac{5}{2}log{|}x+1|-\frac{1}{10}log{|}x-1|-\frac{12}{5}log{|}2x+3|+C }
6. {\displaystyle \int{\frac{1}{x^4-1}dx=\frac{1}{4}log{\left|\frac{x-1}{x+1}\right|}}-\frac{1}{2}{tan}^{-1}{x}+C }
7. {\displaystyle \int\frac{dx}{x(x^2+1)}= l o g{|}x|-\frac{1}{2}log{|}x^2+1|+C }
8. {\displaystyle \int\frac{x^4}{(x-1)(x^2+1)}dx=\frac{x^2}{2}+x+\frac{1}{2}log{|}x-1|-\frac{1}{4}log{|}x^2+1|-\frac{1}{2}{tan}^{-1}{x}+C }
9. {\displaystyle \int\frac{3x^2dx}{x^6+1}={tan}^{-1}{x^3}+C }
10. {\displaystyle \int\frac{x^2}{1-x^6}dx=\frac{1}{6}log{\left|\frac{1+x^3}{1-x^3}\right|}+C }
11. {\displaystyle \int{\frac{x^2}{\sqrt{x^6+a^6}}dx}=\frac{1}{3}log{\left|x^3+\sqrt{x^6+a^6}\right|}+C }
12. {\displaystyle \int\frac{x^3}{\sqrt{1-x^8}}dx=\frac{1}{4}{sin}^{-1}{(}x^4)+C }
13. {\displaystyle \int\frac{x+2}{\sqrt{x^2-1}}dx=\sqrt{x^2-1}+2log{\left|x+\sqrt{x^2-1}\right|}+C }
14. {\displaystyle \int\sqrt{1-\frac{x^2}{9}}dx=\frac{1}{3}\left[\frac{x}{2}\sqrt{9-x^2}+\frac{9}{2}{sin}^{-1}{\frac{x}{3}}\right]+C }
15. {\displaystyle \int\frac{dx}{9x^2+6x+5}=\frac{1}{6}{tan}^{-1}{\left(\frac{3x+1}{2}\right)}+C }
16. {\displaystyle \int\frac{dx}{\sqrt{5x^2-2x}}=\frac{1}{\sqrt5}log{\left|x-\frac{1}{5}+\sqrt{x^2-\frac{2x}{5}}\right|}+C }
17. {\displaystyle \int\frac{dx}{\sqrt{9x-4x^2}}=\frac{1}{2}{sin}^{-1}{\left(\frac{8x-9}{9}\right)}+C }
18. {\displaystyle \int{\sqrt{1-4x-x^2}dx}=\frac{5}{2}{sin}^{-1}{\left(\frac{x+2}{\sqrt5}\right)}+\frac{x+2}{2}\sqrt{1-4x-x^2}+C }
19. {\displaystyle \int{\frac{1}{\sqrt{(x-a)(x-b)}}dx= l o g{\left|x-\frac{a-b}{2}+\sqrt{(x-a)(x-b)}\right|}}+C }
20. {\displaystyle \int\frac{x+2}{2x^2+6x+5}dx=\frac{1}{4}log{|}2x^2+6x+5|+\frac{1}{2}{tan}^{-1}{(}2x+3)+C }
21. {\displaystyle \int{\frac{x+3}{5-4x-x^2}dx=-\sqrt{5-4x-x^2}+{sin}^{-1}{\frac{x+2}{3}}}+C }
22. {\displaystyle \int{\frac{5x+3}{\sqrt{x^2+4x+10}}dx=5\sqrt{x^2+4x+10}-7log{\left|x+2+\sqrt{x^2+4x+10}\right|}}+C }
23. {\displaystyle \int\frac{6x+7}{\sqrt{(x-5)(x-4)}}dx=6\sqrt{x^2-9x+20}+34log{\left|x-\frac{9}{2}+\sqrt{x^2-9x+20}\right|}+C }
24. {\displaystyle \int{(3x-2)\sqrt{x^2+x+1}dx}=(x^2+x+1)^\frac{3}{2}-\frac{7}{2}\left[\left(x+\frac{1}{2}\right)\sqrt{x^2+x+1}+\frac{3}{8}log{\left|x+\frac{1}{2}+\sqrt{x^2+x+1}\right|}\right]+C }
25. {\displaystyle \int{(x-5)\sqrt{x^2+x}}dx=\frac{1}{3}(x^2+x)^\frac{3}{2}-\frac{11}{2}\left\{\frac{2x+1}{4}\sqrt{x^2+x}-\frac{1}{8}log{\left|\left(x+\frac{1}{2}\right)+\sqrt{x^2+x}\right|}\right\}+C }
26. {\displaystyle \int\frac{1}{x(x^n+1)}dx=\frac{1}{n}log{\left|\frac{x^n}{x^n+1}\right|}+C }
27. {\displaystyle \int\frac{dx}{x(x^4-1)}=\frac{1}{4}log{\left|\frac{x^4-1}{x^4}\right|}+C }
28. {\displaystyle \int\frac{1}{(x^2+1)(x^2+4)}dx=\frac{1}{3}{tan}^{-1}{x}-\frac{1}{6}{tan}^{-1}{\frac{x}{2}}+C }
29. {\displaystyle \int\frac{(x^2+1)(x^2+2)}{(x^2+1)(x^2+4)}dx= x+\frac{2}{\sqrt3}{tan}^{-1}{\frac{x}{\sqrt3}}-3{tan}^{-1}{\frac{x}{2}}+C }
30. {\displaystyle \int{\frac{x^2}{(x^2+1)(x^2+4)}dx=-\frac{1}{3}{tan}^{-1}{x}+\frac{2}{3}{tan}^{-1}{\frac{x}{2}}}+C }
31. {\displaystyle \int\frac{cos{x}dx}{(1-sin{x})(2-sin{x})}= l o g{\left|\frac{2-sin{x}}{1-sin{x}}\right|}+C }
32. {\displaystyle \int\frac{1}{(x[6(\log x)^2+7 \log x+2])} dx=\log | \frac{2 \log x+1}{3 \log x+2}|+C }
33. {\displaystyle \int{\frac{e^x}{(e^x+1)(2+e^x)}dx= l o g{\left(\frac{1+e^x}{2+e^x}\right)}}+C }
34. {\displaystyle \int\frac{2x}{(x^2+1)(x^2+3)}dx=\frac{1}{2}log{\left|\frac{x^2+1}{x^2+3}\right|}+C }
35. {\displaystyle \int\frac{(3sin{\theta}-2)cos{\theta}}{5-{cos}^2{\theta}-4sin{\theta}}d\theta=3log{|}2-sin{\theta}|+\frac{4}{2-sin{\theta}}+C }
36. {\displaystyle \int{\frac{e^x}{\sqrt{5-4e^x-e^{2x}}}dx}={sin}^{-1}{\left(\frac{e^x+2}{3}\right)}+C }

### ASSIGNMENT – 5

1. {\displaystyle \int\frac{1}{sin{x}+sin{2}x}dx=\frac{1}{6}log{|}1-cos{x}|+\frac{1}{2}log{|}1+cos{x}|-\frac{2}{3}log{|}1+2cos{x}|+C }
2. {\displaystyle \int{\frac{sin{x}}{sin{4}x}dx=-\frac{1}{8}log{\left|\frac{1+sin{x}}{1-sin{x}}\right|}}+\frac{1}{4\sqrt2}log{\left|\frac{1+\sqrt2sin{x}}{1-\sqrt2sin{x}}\right|}+C }
3. {\displaystyle \int\frac{(x^4-x)^\frac{1}{4}}{x^5}dx=\frac{4}{15}\left(1-\frac{1}{x^3}\right)^\frac{5}{4}+C }
4. {\displaystyle \int{\frac{1}{x\sqrt{ax-x^2}}dx=-\frac{2}{a}\sqrt{\frac{(a-x)}{x}}}+C }
5. {\displaystyle \int_{\frac{1}{3}}^{1}\frac{(x-x^3)^\frac{1}{3}}{x^4}dx=6 }
6. {\displaystyle \int {\frac{1}{x^2(x^4+1)^\frac{3}{4}}}dx=-\left(1+\frac{1}{x^4}\right)^\frac{1}{4}+C }
7. {\displaystyle \int\frac{\sqrt{x^2+1}(log{(}x^2+1)-2log{x})}{x^4}dx=-\frac{1}{3}\left(1+\frac{1}{x^2}\right)^\frac{3}{2}\left[log{\left(1+\frac{1}{x^2}\right)}-\frac{2}{3}\right]+C }
8.   {\displaystyle \int\frac{1}{\sqrt{{cos}^3{x}.cos{(}x+a)}}dx=-2cosec{a}\sqrt{cos{a}-tan{x}.sin{a}}+C }
9. {\displaystyle \int\frac{1}{\sqrt{{sin}^3{x}sin{(}x+\alpha)}}dx=\frac{-2}{sin{\alpha}}\sqrt{\frac{sin{(}x+\alpha)}{sin{x}}}+C }
10. {\displaystyle \int\frac{x^2+4}{x^4+16}dx=\frac{1}{2\sqrt2}{tan}^{-1}{\left(\frac{x^2-4}{2\sqrt2x}\right)}+C }
11. {\displaystyle \int{\sqrt{tan{x}}dx}=\frac{1}{\sqrt2}{tan}^{-1}{\left(\frac{tan{x}-1}{\sqrt{2tan{x}}}\right)}+\frac{1}{2\sqrt2}log{\left|\frac{tan{x}-\sqrt{2tan{x}}+1}{tan{x}+\sqrt{2tan{x}}+1}\right|}+C }
12. {\displaystyle \int\frac{x^2+1}{x^4+x^2+1}dx=\frac{1}{\sqrt3}{tan}^{-1}{\left(\frac{x^2-1}{\sqrt{3x}}\right)}+C }
13. {\displaystyle \int\sqrt{cot{x}}+\sqrt{tan{x}}dx=\sqrt2{tan}^{-1}{\left(\frac{tan{x}-1}{\sqrt{2tan{x}}}\right)}+C }
14. {\displaystyle \int\frac{sin{x}+cos{x}}{\sqrt{sin{2}x}}dx={sin}^{-1}{(}sin{x}-cos{x})+C }
15. {\displaystyle \int_{0}^{\frac{\pi}{4}}\frac{sin{x}+cos{x}}{9+16sin{2}x}dx=\frac{1}{40}log{9} }
16. {\displaystyle \int\left\{\frac{1}{log{x}}-\frac{1}{(log{x})^2}\right\}dx=\frac{x}{log{x}}+C }
17. {\displaystyle \int\left[log{(}log{x})+\frac{1}{(log{x})^2}\right]dx= xlog{(}log{x})-\frac{x}{log{x}}+C }
18. {\displaystyle \int{\frac{1}{\sqrt3sin{x}+cos{x}}dx=\frac{1}{2}log{\left| \tan{\left(\frac{x}{2}+\frac{\pi}{12}\right)}\right|}}+C }
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