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

# Lecture 4 Part 8 Integrals Class 12 Maths

SPECIAL CASES FOR ALL THREE METHODS

Play Video

Questions discussed in this video:

31. $\int \frac{\cos{x} dx}{(1-\sin{x})(2-\sin{x})}$

32. $\int \frac{1}{[6(\log x)^2+7\log x+2]}dx$

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

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

35. $\int \frac{(3\sin{\theta}-2)\cos{\theta}}{5-{\cos}^2 {\theta}-4\sin{\theta}}d\theta$