Find the value of the integral: int frac{2x^2 | vedclass

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Question

(A) Let $I = \int \frac{2x^2}{x^3+1} dx$.
Substitute $u = x^3+1$,then $du = 3x^2 dx$,which implies $x^2 dx = \frac{1}{3} du$.
Substituting these int

Vedclass · Accepted Answer

$\frac{2}{3} \ln |x^3+1| + C$

Vedclass · Answer

(A) Let $I = \int \frac{2x^2}{x^3+1} dx$.
Substitute $u = x^3+1$,then $du = 3x^2 dx$,which implies $x^2 dx = \frac{1}{3} du$.
Substituting these into the integral:
$I = \int \frac{2}{u} \cdot \frac{1}{3} du = \frac{2}{3} \int \frac{1}{u} du$.
$I = \frac{2}{3} \ln |u| + C$.
Substituting $u = x^3+1$ back,we get $I = \frac{2}{3} \ln |x^3+1| + C$.

Find the value of the integral: $\int \frac{2x^2}{x^3+1} dx$

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