The value of int frac{x^{2} d x}{sqrt{x^{6}+a | vedclass

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Question

(C) Let $I = \int \frac{x^{2}}{\sqrt{x^{6}+a^{6}}} dx$. Substitute $x^{3} = t$. Then,$3x^{2} dx = dt$,which implies $x^{2} dx = \frac{1}{3} dt$. Subst

Vedclass · Accepted Answer

$\frac{1}{3} \log \left|x^{3}+\sqrt{x^{6}+a^{6}}\right|+C$

Vedclass · Answer

(C) Let $I = \int \frac{x^{2}}{\sqrt{x^{6}+a^{6}}} dx$. Substitute $x^{3} = t$. Then,$3x^{2} dx = dt$,which implies $x^{2} dx = \frac{1}{3} dt$. Substituting these into the integral,we get: $I = \frac{1}{3} \int \frac{1}{\sqrt{t^{2} + (a^{3})^{2}}} dt$. Using the standard integral formula $\int \frac{1}{\sqrt{x^{2} + a^{2}}} dx = \log |x + \sqrt{x^{2} + a^{2}}| + C$,we have: $I = \frac{1}{3} \log |t + \sqrt{t^{2} + (a^{3})^{2}}| + C$. Substituting $t = x^{3}$ back into the expression: $I = \frac{1}{3} \log |x^{3} + \sqrt{x^{6} + a^{6}}| + C$.

The value of $\int \frac{x^{2} d x}{\sqrt{x^{6}+a^{6}}}$ is equal to

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