# VIDEO - 9

Questions based on Particular Cases

9. $\int \frac{3x^2dx}{x^6+1}={\tan}^{-1}{x^3}+C$

10. $\int \frac{x^2}{1-x^6}dx=\frac{1}{6} \log {\left|\frac{1+x^3}{1-x^3}\right|}+C$

11. $\int \frac{x^2}{\sqrt{x^6+a^6}} dx = \frac{1}{3} \log \left | x^3 + \sqrt{x^6+a^6} \right | +C$

12. $\int \frac{x^3}{\sqrt{1-x^8}}dx=\frac{1}{4}{\sin}^{-1}{(}x^4)+C$

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

14. $\int \sqrt{1-\frac{x^2}{9}}dx=\frac{1}{3}\left[\frac{x}{2}\sqrt{9-x^2}+\frac{9}{2}{\sin}^{-1}{\frac{x}{3}}\right]+C$

# VIDEO - 8

INTEGRATION OF ALGEBRAIC FUNCTIONS

SECOND METHOD

In the second method, we use integrals of some particular functions to find the integrals of our rational functions.

Some tips to remember PARTICULAR CASES

$$\int \frac{dx}{x^4-1}$$

# VIDEO - 7

INTEGRATION OF ALGEBRAIC FUNCTIONS

SECOND METHOD

In the second method, we use integrals of some particular functions to find the integrals of our rational functions.

Derivation of PARTICULAR CASES

$\int \sqrt{x^2\pm\ a^2} = \frac{x}{2}\sqrt{x^2\pm\ a^2} \pm \frac{a^2}{2} \log\left| x+ \sqrt {x^2\pm\ a^2}\right|+C$

$\int \sqrt{a^2-x^2}v=v\frac{x}{2} \sqrt{a^2-x^2} + \frac{a^2}{2}{sin}^{-1}{\frac{x}{a}}+C$

## VIDEO - 6

INTEGRATION OF ALGEBRAIC FUNCTIONS

SECOND METHOD

In the second method, we use integrals of some particular functions to find the integrals of our rational functions.

Derivation of PARTICULAR CASES

$\int \frac{dx}{\sqrt{x^2\pm a^2}}= \log{\left|x+\sqrt{x^2\pm a^2}\right|}+C$

$\int \frac{dx}{\sqrt{a^2-x^2}}={\sin}^{-1}{\frac{x}{a}}+C$

## VIDEO - 5

INTEGRATION OF ALGEBRAIC FUNCTIONS

SECOND METHOD

In the second method, we use integrals of some particular functions to find the integrals of our rational functions.

Derivation of PARTICULAR CASES

$\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\log{\left|\frac{x-a}{x+a}\right|}+C$

$\int\frac{dx}{x^2+a^2}=\frac{1}{a}{\tan}^{-1}{\frac{x}{a}}+C$

$\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\log{\left|\frac{a+x}{a-x}\right|}+C$

## VIDEO - 4

In this lecture I am discussing following questions and the last case of partial fractions

7. $\int\frac{1}{x(x^2+1)}dx$

8. $\int \frac{x^4}{(x-1)(x^2+1)}dx$

# Lecture 3 Part 1 Integrals Class 12 Maths :::INTEGRATION BY PARTS:::

Method to identify the First Function:

I    L    A    T    E

I = Inverse Trigonometric Functions
L = Logarithmic Functions
A = Algebraic Functions
T = Trigonometric Functions
E = Exponential functions

## VIDEO - 1 CLIP - 2

4. $\int{x\log{2}x}dx=\frac{x^2}{2}\log{2}x-\frac{x^2}{4}+C$

5. $\int{x^5. \log{x}dx=}\frac{x^3}{3}log{x}-\frac{x^3}{9}+C$

6. $\int{x{\tan}^{-1}{x}}dx=\frac{x^2}{2}{\tan}^{-1}{x}-\frac{x}{2}+\frac{1}{2}{\tan}^{-1}{x}+C$

7. $\int(x^2+1)\log{x}dx=\left(\frac{x^3}{3}+x\right)\log{x}-\frac{x^3}{9}-x+C$

8. $\int e^{2x}\sin{x}dx=\frac{e^{2x}}{5}(2\sin{x} - \cos{x})+C$

## Video - 3

In this lecture I am discussing following questions and the last case of partial fractions

3. $\int\frac{x^2+1}{x^2-5x+6}dx$

4. $\int \frac{x^2+x+1}{(x+2)(x^2+1)}dx$

5. $\int\frac{2x-3}{(x^2-1)(2x+3)}dx$

6. $\int\frac{1}{x^4-1}dx$

## Video - 2

In this video, I am discussing second case of partial fractions and the following question:

2. $\int\frac{3x-2}{(x+1)^2(x+3)}dx$

## Inverse Trigonometric Functions Lecture 5 Part 2

Play Video

Questions discussed in this lecture

Solve:
9. $\tan ^{-1} (x-1)+\tan ^{-1} x+\tan ^{-1} (x+1)=\tan ^{-1} 3x$

10. $3\sin ^{-1} \frac{2x}{1+x^{2} } -4\cos ^{-1} \frac{1-x^{2} }{1+x^{2} } +2\tan ^{-1} \frac{2x}{1-x^{2} } =\frac{\pi }{3}$

11. If $\sin ^{-1} \frac{2a}{1+a^{2} } -\cos ^{-1} \frac{1-b^{2} }{1+b^{2} } =\tan ^{-1} \frac{2x}{1-x^{2} }$, then prove that $x=\frac{a-b}{1+ab}.$

12. Evaluate: $\tan ^{-1} \left(\frac{a+bx}{b-ax} \right),\, \, x<\frac{b}{a}$

13. Prove: $\tan ^{-1} \left(\frac{a-b}{1+ab} \right)+\tan ^{-1} \left(\frac{b-c}{1+bc} \right)+\tan ^{-1} \left(\frac{c-a}{1+ca} \right)=0$

14. If $\tan ^{-1} x+\tan ^{-1} y=\frac{4\pi }{5}$, then find the value of $\cot ^{-1} x+\cot ^{-1}y$?

15. If $\tan ^{-1} \left(\frac{1}{1+1.2} \right)+\tan ^{-1} \left(\frac{1}{1+2.3} \right)+…+\tan ^{-1} \left(\frac{1}{1+n.(n+1)} \right)=\tan ^{-1} \phi$, then find the value of $\phi$.

16. If $(\tan ^{-1} x)^{2} +(\cot ^{-1} x)^{2} =\frac{5\pi ^{2} }{8}$, then find x.