# L1 - Direct Integrals

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. Some of the identities are:
$1+\cos x=2\cos ^{2} \frac{x}{2}$

$1-\cos x=2\sin ^{2} \frac{x}{2}$

$1\pm \sin x=\left(\cos \frac{x}{2} \pm \sin \frac{x}{2} \right)^{2}$

$\frac{1+\tan x}{1-\tan x} =\tan \left(\frac{\pi }{4} +x\right)$

$\frac{1-\tan x}{1+\tan x} =\tan \left(\frac{\pi }{4} -x\right)$

$\log e^{x} =x$

$e^{\log x} =x$

Play Video

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

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