Calculus Examples

$\int_{- 2}^{4} x^{2} {(x^{3} + 8)}^{2} d x$

Step 1

Let $u = x^{3} + 8$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{3} + 8$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{3} + 8$ .

$\frac{d}{d x} [x^{3} + 8]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{3} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [8]$

Step 1.1.4

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 1.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{3} + 8$ .

$u_{lower} = {(- 2)}^{3} + 8$

Step 1.3

Simplify.

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Step 1.3.1

Raise $- 2$ to the power of $3$ .

$u_{lower} = - 8 + 8$

Step 1.3.2

Add $- 8$ and $8$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{3} + 8$ .

$u_{upper} = 4^{3} + 8$

Step 1.5

Simplify.

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Step 1.5.1

Raise $4$ to the power of $3$ .

$u_{upper} = 64 + 8$

Step 1.5.2

Add $64$ and $8$ .

$u_{upper} = 72$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 72$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{0}^{72} u^{2} \frac{1}{3} d u$

Step 2

Combine $u^{2}$ and $\frac{1}{3}$ .

$\int_{0}^{72} \frac{u^{2}}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{0}^{72} u^{2} d u$

Step 4

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{3} \frac{1}{3} u^{3}]_{0}^{72}$

Step 5

Simplify the expression.

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Step 5.1

Evaluate $\frac{1}{3} u^{3}$ at $72$ and at $0$ .

$\frac{1}{3} ((\frac{1}{3} \cdot 72^{3}) - \frac{1}{3} \cdot 0^{3})$

Step 5.2

Raise $72$ to the power of $3$ .

$\frac{1}{3} (\frac{1}{3} \cdot 373248 - \frac{1}{3} \cdot 0^{3})$

Step 5.3

Simplify.

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Step 5.3.1

Combine $\frac{1}{3}$ and $373248$ .

$\frac{1}{3} (\frac{373248}{3} - \frac{1}{3} \cdot 0^{3})$

Step 5.3.2

Cancel the common factor of $373248$ and $3$ .

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Step 5.3.2.1

Factor $3$ out of $373248$ .

$\frac{1}{3} (\frac{3 \cdot 124416}{3} - \frac{1}{3} \cdot 0^{3})$

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor $3$ out of $3$ .

$\frac{1}{3} (\frac{3 \cdot 124416}{3 (1)} - \frac{1}{3} \cdot 0^{3})$

Step 5.3.2.2.2

Cancel the common factor.

$\frac{1}{3} (\frac{3 \cdot 124416}{3 \cdot 1} - \frac{1}{3} \cdot 0^{3})$

Step 5.3.2.2.3

Rewrite the expression.

$\frac{1}{3} (\frac{124416}{1} - \frac{1}{3} \cdot 0^{3})$

Step 5.3.2.2.4

Divide $124416$ by $1$ .

$\frac{1}{3} (124416 - \frac{1}{3} \cdot 0^{3})$

Step 5.4

Raising $0$ to any positive power yields $0$ .

$\frac{1}{3} (124416 - \frac{1}{3} \cdot 0)$

Step 5.5

Simplify.

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Step 5.5.1

Multiply $0$ by $- 1$ .

$\frac{1}{3} (124416 + 0 (\frac{1}{3}))$

Step 5.5.2

Multiply $0$ by $\frac{1}{3}$ .

$\frac{1}{3} (124416 + 0)$

Step 5.6

Add $124416$ and $0$ .

$\frac{1}{3} \cdot 124416$

Step 5.7

Simplify.

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Step 5.7.1

Combine $\frac{1}{3}$ and $124416$ .

$\frac{124416}{3}$

Step 5.7.2

Cancel the common factor of $124416$ and $3$ .

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Step 5.7.2.1

Factor $3$ out of $124416$ .

$\frac{3 \cdot 41472}{3}$

Step 5.7.2.2

Cancel the common factors.

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Step 5.7.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 41472}{3 (1)}$

Step 5.7.2.2.2

Cancel the common factor.

$\frac{3 \cdot 41472}{3 \cdot 1}$

Step 5.7.2.2.3

Rewrite the expression.

$\frac{41472}{1}$

Step 5.7.2.2.4

Divide $41472$ by $1$ .

$41472$