x-dx-a-4-x-4-

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

\int \frac{x d x}{\sqrt{a^{4} + x^{4}}}

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

x^{2} = a^{2} \sinh u

2 x d x = a^{2} \cosh u d u

\int \frac{a^{2} \cosh u d u}{2 a^{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

l e t x^{2} = a^{2} \tan θ

\Rightarrow 2 x d x = a^{2} \sec^{2} θ d θ

\int \frac{x d x}{\sqrt{a^{4} + x^{4}}} = \frac{a^{2}}{2} \int \frac{\sec^{2} θ d θ}{a^{2} \sec θ}

= \frac{1}{2} \ln ∣ \sec θ + \tan θ ∣ + c_{1}

= \frac{1}{2} \ln ∣ \frac{\sqrt{x^{2} + a^{2}} + x^{2}}{a^{2}} ∣ + c_{1}

= \frac{1}{2} \ln (x^{2} + \sqrt{x^{2} + a^{2}}) + c .

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