Mathematics

Evaluate $$\displaystyle \int\cfrac{x^{2}}{\sqrt{x^{6}+a^{6}}} \ dx$$


SOLUTION
Let $$I=\displaystyle \int \cfrac{x^{2}}{\sqrt{x^{6}+a^{6}}} dx$$
Put $$x^3=t$$
$$3x^2 dx=dt$$
$$x^2 dt=dt/3$$
$$I=\displaystyle \int \dfrac{x^2}{\sqrt{(x^3)^2+a^6}} dx$$
$$=\dfrac{1}{3} \displaystyle \int \dfrac{dt}{\sqrt{t^2+(a^3)^2}}dx$$
$$=\dfrac{1}{3} \times \log |t+ \sqrt{t^2+(a^3)^2}|+C$$
$$=\displaystyle \dfrac{1}{3} \log |x^3+ \sqrt{x^6+(a^6}|+C$$
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Subjective Medium Published on 17th 09, 2020
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