Lecture 4 Part 2 Integrals Class 12 Maths

Lecture 4 Part 2 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.

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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}{\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