Question

$\int x^4(1+x^5)^3dx$

Answer 1

$\frac{(1+x^5)^4}{20}+C$

Answer 2

Solution:

Let

$$u = 1 + x^5 \quad \Rightarrow \quad \frac{du}{dx} = 5x^4 \quad \Rightarrow \quad x^4\,dx = \frac{du}{5}.$$

Substitute in the integral:

$$\int x^4(1+x^5)^3\,dx = \int (1+x^5)^3 x^4\,dx = \int u^3 \frac{du}{5} = \frac{1}{5}\int u^3\,du.$$

Integrate:

$$\frac{1}{5} \cdot \frac{u^4}{4} = \frac{u^4}{20} + C.$$

Substitute back:

$$\frac{(1+x^5)^4}{20} + C.$$

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Minimal Explanation:
Substitute $u=1+x^5$ so that $du=5x^4dx$. Integrate to get $\frac{u^4}{20}+C$ and revert substitution.