# Ch07. Properties of Integrals

## Achievements

The following badges can be earned while learning.

## Certificate

Certificate on successful completion of this course.

This online course is an extended part of chapter 7 Integrals of Class 12 Maths. In this course, you will learn properties of definite Integrals and their derivations. For further understanding of concepts and for examination preparation, practice questions based on the above topics are discussed in the form of assignments that have questions from NCERT Textbook exercise 7.11, Board’s Question Bank, RD Sharma, NCERT Exemplar etc. instead of only one book. The PDF of assignments can be downloaded within the course.

## Course Content

PDF Files
Concept & Derivation
Assignment - 1
Concept & Derivation
Assignment - 1
Concept & Derivation
Assignment - 2
Concept & Derivation
Assignment - 2

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

Question 1. {\displaystyle \int_{2}^{8}{\,|x-5|}\,dx=9}
Question 2. {\displaystyle \int_{-1}^{2}{|{{x}^{3}}-x|\,dx=\frac{11}{4}}}
Question 3. {\displaystyle \int_{-1}^{\frac{3}{2}}{\left| x.sin(\pi x) \right|}\,dx=\frac{3}{\pi }+\frac{1}{{{\pi }^{2}}}}
Question 4. {\displaystyle \int_{\frac{\pi }{6}}^{\frac{\pi }{3}}{\frac{sinx-cosx}{\sqrt{sin2x}}\,dx=0}}
Question 5. {\displaystyle \int_{\frac{\pi }{6}}^{\frac{\pi }{3}}{\frac{dx}{1+\sqrt{tanx}}}=\frac{\pi }{12}}
Question 6. {\displaystyle \int_{0}^{\frac{\pi }{2}}{co{{s}^{2}}x\,dx}=\frac{\pi }{4}}
Question 7. {\displaystyle \int_{0}^{2}{x\sqrt{2-x}}\,dx=\frac{16\sqrt{2}}{15}}
Question 8. {\displaystyle \int_{0}^{1}{x{{(1-x)}^{n}}\,dx=}\frac{1}{(n+1)(n+2)}}
Question 9. {\displaystyle \int_{0}^{\frac{\pi }{2}}{\frac{si{{n}^{\frac{3}{2}}}x\,dx}{si{{n}^{\frac{3}{2}}}x+co{{s}^{\frac{3}{2}}}x}}=\frac{\pi }{4}}
Question 10. {\displaystyle \int_{0}^{a}{\frac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx}=\frac{a}{2}}
Question 11. {\displaystyle \int_{0}^{\frac{\pi }{4}}{log(1+tanx)\,dx}=\frac{\pi }{8}log2}
Question 12. {\displaystyle \int_{0}^{\frac{\pi }{2}}{\frac{sinx-cosx}{1+sinxcosx}\,dx=0}}
Question 13. {\displaystyle \int_{0}^{\pi }{\frac{xsinx}{1+co{{s}^{2}}x}dx=\frac{{{\pi }^{2}}}{4}}}
Question 14. {\displaystyle \int_{0}^{\pi }{\frac{x}{1+sinx}\,dx=\pi }}
Question 15. {\displaystyle \int_{0}^{\pi }{\frac{xtanx}{secx+tanx}\,dx=\frac{\pi }{2}(\pi -2)}}
Question 16. {\displaystyle \int_{0}^{\frac{\pi }{2}}{(2logsinx-logsin2x)\,dx=\frac{\pi }{2}log\frac{1}{2}}}

### Assignment – 2

Question 1. {\displaystyle \int_{0}^{2\pi }{co{{s}^{5}}x\,dx}=0}
Question 2. {\displaystyle \int_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{si{{n}^{2}}x.dx}=\frac{\pi }{2}}
Question 3. {\displaystyle \int_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{si{{n}^{7}}x.dx=0}}
Question 4. {\displaystyle \int_{-1}^{1}{{{x}^{17}}co{{s}^{4}}x\,dx=0}}
Question 5. {\displaystyle \int_{-1}^{1}{si{{n}^{5}}xco{{s}^{4}}x\,dx=0}}
Question 6. {\displaystyle \int\limits_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{log\left( \frac{2-sinx}{2+sinx} \right)}\,dx=\,0}
Question 7. {\displaystyle \int_{\frac{-\pi }{2}}^{\frac{\pi }{2}}{({{x}^{3}}+xcosx+ta{{n}^{5}}x+1)\,dx=\pi }}
Question 8. {\displaystyle \int_{0}^{\frac{\pi }{2}}{log\left( \frac{4+3sinx}{4+3cosx} \right)}\,dx=0}
Question 9. {\displaystyle \int_{0}^{\frac{\pi }{2}}{sin2xta{{n}^{-1}}(sinx)\,dx=\frac{\pi }{2}-1}}
Question 10. If {\displaystyle f(a+b-x)=f(x)} then show that {\displaystyle \int_{a}^{b}{x\,f(x)dx}=\frac{a+b}{2}\int_{a}^{b}{f(x)\,dx}} .
Question 11. Show that {\displaystyle \int_{0}^{a}{f(x)g(x)\,dx=2\int_{0}^{a}{f(x)\,dx}}} , if f and g are defined as {\displaystyle f(x)=f(a-x)} and {\displaystyle g(x)+g(a-x)=4} .
Question 12. {\displaystyle {{\int_{{}}^{{}}{{{e}^{x}}\left( \frac{1-x}{1+{{x}^{2}}} \right)}}^{2}}\,dx=\frac{{{e}^{x}}}{1+{{x}^{2}}}+C}
Question 13. {\displaystyle \int_{{}}^{{}}{{{e}^{ta{{n}^{-1}}x}}\left( \frac{1+x+{{x}^{2}}}{1+{{x}^{2}}} \right)}\,dx=x{{e}^{ta{{n}^{-1}}x}}+C}
Question 14. {\displaystyle \int_{{}}^{{}}{si{{n}^{-1}}\sqrt{\frac{x}{a+x}}\,dx\,\,\,\,\,\,=\,\,\,\,\,a\left[ \frac{x}{a}ta{{n}^{-1}}\sqrt{\frac{x}{a}}-\sqrt{\frac{x}{a}}+ta{{n}^{-1}}\sqrt{\frac{x}{a}} \right]}+C}
Question 15. {\displaystyle \int_{0}^{\pi }{\frac{xdx}{{{a}^{2}}co{{s}^{2}}x+{{b}^{2}}si{{n}^{2}}x}=\frac{{{\pi }^{2}}}{2ab}}}
Question 16. {\displaystyle \int_{0}^{\pi }{log(1+cosx)\,dx=-\pi log2}}

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