Math / Calculus

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

Studdy Solution

STEP 1

1. We are given the integral $\int x^{5} \sqrt{x^{3}+1} \, dx$ .
2. We will use substitution to simplify the integral.

STEP 2

1. Choose a substitution to simplify the integral.
2. Differentiate the substitution to find $dx$ .
3. Substitute into the integral and simplify.
4. Integrate the simplified expression.
5. Substitute back to the original variable.

STEP 3

Choose a substitution. Let $u = x^3 + 1$ .

STEP 4

Differentiate the substitution to find $dx$ :
$\frac{du}{dx} = 3x^2$ $du = 3x^2 \, dx$ $dx = \frac{du}{3x^2}$

STEP 5

Substitute $u = x^3 + 1$ and $dx = \frac{du}{3x^2}$ into the integral:
$\int x^{5} \sqrt{x^{3}+1} \, dx = \int x^{5} \sqrt{u} \cdot \frac{du}{3x^2}$
Simplify the expression:
$= \frac{1}{3} \int x^{3} \sqrt{u} \, du$
Since $x^3 = u - 1$ (from $u = x^3 + 1$ ), substitute $x^3$ :
$= \frac{1}{3} \int (u - 1) \sqrt{u} \, du$

STEP 6

Distribute and integrate the expression:
$= \frac{1}{3} \int (u^{3/2} - u^{1/2}) \, du$
Integrate term by term:
$= \frac{1}{3} \left( \frac{2}{5} u^{5/2} - \frac{2}{3} u^{3/2} \right) + C$
$= \frac{1}{3} \left( \frac{2}{5} u^{5/2} \right) - \frac{1}{3} \left( \frac{2}{3} u^{3/2} \right) + C$
$= \frac{2}{15} u^{5/2} - \frac{2}{9} u^{3/2} + C$

STEP 7

Substitute back to the original variable $x$ :
$= \frac{2}{15} (x^3 + 1)^{5/2} - \frac{2}{9} (x^3 + 1)^{3/2} + C$
The solution to the integral is:
$\boxed{\frac{2}{15} (x^3 + 1)^{5/2} - \frac{2}{9} (x^3 + 1)^{3/2} + C}$

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