Lecture 4 Integrals Class 12 Maths Part 9

Integrals Class 12

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

Lecture 4 Integrals Class 12 Maths Part 8

Integrals Class 12

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}

Part 7 Lecture 4 Integrals Class 12 Maths

Integrals Class 12

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

Integrals Class 12 Maths Part 6 Lecture 4

Integrals Class 12

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

Part 5 Lecture 4 Integrals Class 12 Maths

Integrals Class 12

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

 

Lecture 4 Part 4 Integrals Class 12 Maths

Integrals Class 12

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 Clip 2 Integrals Class 12 Maths

Integrals Class 12

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

Lecture 3 Part 1 Clip 2 Integrals Class 12 Maths 1

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

Lecture 4 Part 3 Integrals Class 12 Maths

Integrals Class 12

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

Lecture 4 Part 2 Integrals Class 12 Maths

Integrals Class 12

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

2 Inverse Trigonometry Functions Class 12 maths
inverse_trigonometric_functions_lecture_5_part_2
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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.