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intx^(5)sqrt(x^(3)+1)dx

$\int x^{5} \sqrt{x^{3}+1} d x$

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Evaluate definite integrals using the chain rule

Full solution

Q.

\int x^{5} \sqrt{x^{3}+1} d x

Substitute $u = x^3 + 1$ : Let's start by substituting $u = x^3 + 1$ , then $du = 3x^2 dx$ .
Find $dx$ : Rearrange to find $dx$ : $dx = \frac{du}{3x^2}$ .
Substitute into the integral: Substitute into the integral: $\int x^5 \sqrt{x^3 + 1} \, dx = \int x^5 \sqrt{u} \left(\frac{du}{3x^2}\right)$ .
Simplify the integral: Simplify the integral: $\int x^3 \sqrt{u} \left(\frac{1}{3}\right) du$ .
Rewrite in terms of $u$ : Rewrite $x^3$ in terms of $u$ : $x^3 = u - 1$ , so the integral becomes $\int (u - 1) \cdot \sqrt{u} \cdot \left(\frac{1}{3}\right) \, du$ .
Expand and simplify: Expand and simplify: $(\frac{1}{3}) \times \int (u^{\frac{3}{2}} - u^{\frac{1}{2}}) \, du$ .
Integrate term by term: Integrate term by term: $\frac{1}{3}$ * $\left[\frac{2}{5} * u^{\frac{5}{2}} - \frac{2}{3} * u^{\frac{3}{2}}\right]$ + $C$ .

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