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Question Number 43417 by Raj Singh last updated on 10/Sep/18

Commented by alex041103 last updated on 10/Sep/18

$$isthat\sqrt[x]{x^n+a^n}$$

Commented by maxmathsup by imad last updated on 10/Sep/18

$$let{I}_n=\int \frac{dx}{x\sqrt{x^n+a^n}}\Rightarrow I_n=\int \frac{dx}{x{a}^{\frac{n}{2}}\sqrt{{\left(\frac{x}{a}\right)}^n+1}}changement{\left(\frac{x}{a}\right)}^n={tan}^2\theta$$$$give\frac{x}{a}={\left({tan}^2\theta \right)}^{\frac{1}{n}}\Rightarrow dx=\frac{a}{n}2tan\left(\theta \right)(1+{tan}^2\theta ){\left({tan}^2\theta \right)}^{\frac{1}{n}-1}d\theta$$$$I_n=\frac{2a}{n}\int \frac{tan\theta (1+{tan}^2\theta ){\left({tan}^2\theta \right)}^{\frac{1}{n}-1}}{a{\left({tan}^2\theta \right)}^{\frac{1}{n}}\sqrt{1+{tan}^2\theta }}d\theta$$$$=\frac{2}{n}\int {tan}^{-1}\theta \sqrt{1+{tan}^2\theta }d\theta =\frac{2}{n}\int \frac{1}{{cos}^2\theta \frac{sin\theta }{cos\theta }}d\theta$$$$=\frac{2}{n}\int \frac{2d\theta }{sin\left(2\theta \right)}{=}_{2\theta =u}\frac{2}{n}\int \frac{du}{sinu}{=}_{tan\left(\frac{u}{2}\right)=\alpha }\frac{2}{n}\int \frac{1}{\frac{2\alpha }{1+{\alpha }^2}}\frac{2d\alpha }{1+{\alpha }^2}$$$$=\frac{2}{n}\int \frac{d\alpha }{\alpha }=\frac{2}{n}ln\mid \alpha \mid +c=\frac{2}{n}ln\mid tan\left(\frac{u}{2}\right)\mid =\frac{2}{n}ln\mid tan\theta \mid$$$$=\frac{2}{n}ln\mid tan\left(arctan{\left(\frac{x}{a}\right)}^{\frac{n}{2}}\right)\mid +c.$$$$$$

Answered by tanmay.chaudhury50@gmail.com last updated on 10/Sep/18

$$\int \frac{dx}{x\sqrt{x^n+a^n}}$$$$\int \frac{x^{n-1}dx}{x^n\sqrt{x^n+a^n}}$$$$t^2=x^n+a^n$$$$2tdt={nx}^{n-1}dx$$$$\int \frac{2tdt}{n\times (t^2-a^n)\times t}$$$$\frac{2}{n}\int \frac{dt}{t^2-{\left(a^{\frac{n}{2}}\right)}^2}$$$$\frac{2}{n}\times \frac{1}{2a^{\frac{n}{2}}}\int \frac{(t+a^{\frac{n}{2}})-(t-a^{\frac{n}{2}})}{(t+a^{\frac{n}{2}})(t-a^{\frac{n}{2}})}dt$$$$=\frac{1}{{na}^{\frac{n}{2}}}[\int \frac{dt}{t-a^{\frac{n}{2}}}-\int \frac{dt}{t+a^{\frac{n}{2}}}]$$$$=\frac{1}{{na}^{\frac{n}{2}}}ln\mid \frac{t-a^{\frac{n}{2}}}{t+a^{\frac{n}{2}}}\mid +c$$$$\frac{1}{{na}^{\frac{n}{2}}}ln\mid \frac{\sqrt{x^n+a^n}-a^{\frac{n}{2}}}{\sqrt{x^n+a^n}+a^{\frac{n}{2}}}\mid +c$$

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