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Question Number 25290 by rather ishfaq last updated on 07/Dec/17

$$\int \frac{xdx}{\sqrt{a^4+x^4}}$$

Answered by prakash jain last updated on 07/Dec/17

$$x^2=a^2\sinh u$$$$2xdx=a^2\cosh udu$$$$\int \frac{a^2\cosh udu}{2a^2\cosh u}$$$$=\frac{1}{2}u+C$$$$=\frac{1}{2}{\sinh }^{-1}\frac{x^2}{a^2}+C$$

Answered by ajfour last updated on 07/Dec/17

$$let{x}^2=a^2\tan \theta$$$$\Rightarrow 2xdx=a^2\sec {}^2\theta d\theta$$$$\int \frac{xdx}{\sqrt{a^4+x^4}}=\frac{a^2}{2}\int \frac{\sec {}^2\theta d\theta }{a^2\sec \theta }$$$$=\frac{1}{2}\ln \mid \sec \theta +\tan \theta \mid +c_1$$$$=\frac{1}{2}\ln \mid \frac{\sqrt{x^2+a^2}+x^2}{a^2}\mid +c_1$$$$=\frac{1}{2}\ln (x^2+\sqrt{x^2+a^2})+c.$$

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