Lecture 4 Part 3 Integrals Class 12 Maths

Lecture 4 Part 3 Integrals Class 12 Maths

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.

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

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

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

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

These identities can be applied only when
    There is no function in the numerator. If there is any function in numerator try to remove that with the help of substitution or simplification.
    The variable part must be in x^2 \text{ or } (x\pm\ n)^2 form. There’s no restriction with the constant part.

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Questions discussed in this video:

9. \int\frac{3x^2dx}{x^6+1}

10. \int\frac{x^2}{1-x^6}dx

11. \int\frac{x^2}{\sqrt{x^6+a^6}}dx

12. \int\frac{x^3}{\sqrt{1-x^8}}dx

13. \int\frac{x+2}{\sqrt{x^2-1}}dx

14. \int\sqrt{1-\frac{x^2}{9}}dx