Question

\int \left(x^3+1\right)^5x^2dx

Answer 1

$\int \left(x^3+1\right)^5x^2dx$

Here, we apply substitution method I.e.

Let $u = x^3 + 1$

$\implies \frac{du}{dx} = 3x^2$

$\implies dx= \frac{1}{3x^2}du$

Replacing back, we have;

$\int (u)^5 \cancel{x^2} \frac{1}{3 \cancel{x^2} }du$ $= \frac{1}{3} \int u^5 du$

Now we solve;

$\int u^5 du$

Here, we apply power rule;

Where we let;

$\int u^n du = \frac{u^{n+1}}{n+1}$ With $n = 5$ , we have;

$= \frac{u^6}{6}$

Replacing this back to $\frac{1}{3} \int u^5 du$ we have;

$\frac{u^6}{18}$

Now we undo the substitution

$u = x^3 +1$

$\frac{(x^3 +1)^6}{18}$ and we add a C to this, hence the final answer i.e.

$\frac{(x^3 +1)^6}{18} + C$