Mathematics

Solve : $$\displaystyle \int \dfrac{x \, dx }{(x^2 + a^2) (x^2 + b^2)}$$


SOLUTION
$$\int \dfrac{x}{(x^2+a^2)(x^2+b^2)}dx$$

substitute $$u=\dfrac{x^2}{2}\rightarrow dx=\dfrac{1}{x}du$$

$$=\int \dfrac{1}{(2u+a^2)(2u+b^2)}du$$

Upon partial fraction decomposition

$$=\int \left ( \dfrac{1}{(b^2-a^2)(2u+a^2)}-\dfrac{1}{(b^2-a^2)(2u+b^2)} \right )du$$

upon linearity

$$=\dfrac{1}{(b^2-a^2)}\int \dfrac{1}{2u+a^2}du-\dfrac{1}{(b^2-a^2)}\int \dfrac{1}{2u+b^2}du$$

$$=\dfrac{1}{(b^2-a^2)}\dfrac{\ln (2u+a^2)}{2}-\dfrac{1}{(b^2-a^2)}\dfrac{\ln (2u+b^2)}{2}$$

$$=-\dfrac{\ln(x^2+b^2)-\ln (x^2+a^2) }{2(b^2-a^2)}+C$$
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Subjective Medium Published on 17th 09, 2020
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