Lecture 1 Part 1 Integrals Class 12 Maths

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


Integrals Lecture 1 Part 1
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

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|