भीड़ हमेशा उस रास्ते पर चलती है जो रास्ता आसान लगता है,

लेकिन इसका मतलब यह नहीं की भीड़ हमेशा सही रास्ते पर चलती है|

अपने रास्ते खुद चुनिए क्योंकि आपको आपसे बेहतर और कोई नहीं जानता|

**Logics for this lecture:**

*Question must be in addition/subtraction form.*

*Try to convert division/multiplication between functions to addition/subtraction by simplification.*

*If there are functions other than algebraic functions like trigonometry and logarithmic function, then you can use their respective identities to simplify them.*

**Logics for this lecture:**

*Question must be in addition/subtraction form.*

*Try to convert division/multiplication between functions to addition/subtraction by simplification.*

*If there are functions other than algebraic functions like trigonometry and logarithmic function, then you can use their respective identities to simplify them.*

Questions Discussed in this lecture:

1. \(\int x^{2} \left(1-\frac{1}{x^{2} } \right), dx =\frac{x^{3} }{3} -x+C\)

2. \(\int _{}^{}\left(\sqrt{x} -\frac{1}{\sqrt{x} } \right) ^{2} dx=\frac{x^{2} }{2} +\log |x|-2x+C\)

3. \(\int \left(8^{x} +x^{8} +\frac{8}{x} +\frac{x}{8} \right), dx =\frac{8^{x} }{\log 8} +\frac{x^{9} }{9} +8\log |x|+\frac{x^{2} }{16} +C\)

4. \(\int (e^{a\log x} +e^{x\log a} ), dx =\frac{x^{a+1} }{a+1} +\frac{a^{x} }{\log a} +C\)

5. \(\int \left(\frac{\cos 2x+2\sin ^{2} x}{\cos ^{2} x} \right) , dx=\tan x+C\)

6. \(\int (x^{c} +c^{x} ), dx =\frac{x^{c+1} }{c+1} +\frac{c^{x} }{\log c} +C\)

7. \(\int _{}^{}\frac{x^{2} +3x+4}{\sqrt{x} } , dx=\frac{2}{5} x^{\frac{5}{2} } +2x^{\frac{3}{2} } +8\sqrt{x} +C\)

8. \(\int \left(\frac{2a}{\sqrt{x} } -\frac{b}{x^{2} } +3c, , \sqrt[{3}]{x^{2} } \right) , dx=4a\sqrt{x} +\frac{b}{x} +\frac{9cx^{\frac{5}{3} } }{5} +C\)

9. \(\int _{}^{}\frac{x^{3} -x^{2} +x-1}{x-1} , , dx =\frac{x^{3} }{3} +x+C\)

10. \(\int _{}^{}\frac{sec ^{2} x}{{\rm \cosec}^{2} x} , dx=\tan x-x+C\)

11. \(\int \sqrt{1+\sin 2x} , dx=-\cos x+\sin x+C\)