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Question Number 205245 by pticantor last updated on 13/Mar/24

$$\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}$$$$$$$$\int \frac{3\mathit{x}\mathit{d}\mathit{x}}{\sqrt{1-{\mathit{x}}^4}}$$$$\mathit{p}\mathit{l}\mathit{s}\mathit{s}\mathit{s}\mathit{s}\mathit{s}\mathit{s}\mathit{s}$$

Answered by mr W last updated on 13/Mar/24

$$=\frac{3}{2}\int \frac{d\left(x^2\right)}{\sqrt{1-{\left(x^2\right)}^2}}$$$$=\frac{3}{2}\int \frac{du}{\sqrt{1-u^2}}withu=x^2$$$$=\frac{3}{2}\int \frac{d\left(\sin \theta \right)}{\sqrt{1-{\sin }^2\theta }}withu=\sin \theta$$$$=\frac{3}{2}\int \frac{\cos \theta d\theta }{\cos \theta }$$$$=\frac{3}{2}\int d\theta$$$$=\frac{3}{2}\theta +C$$$$=\frac{3}{2}{\sin }^{-1}u+C$$$$=\frac{3}{2}{\sin }^{-1}{x}^2+C$$

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