Calculus Examples

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

Step 1

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

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Step 1.1

Let $u = x^{2} + 2$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 2$ .

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

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2]$ .

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

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [2]$

Step 1.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{2} + 2$ .

$u_{lower} = 2^{2} + 2$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 4 + 2$

Step 1.3.2

Add $4$ and $2$ .

$u_{lower} = 6$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{2} + 2$ .

$u_{upper} = 4^{2} + 2$

Step 1.5

Simplify.

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Step 1.5.1

Raise $4$ to the power of $2$ .

$u_{upper} = 16 + 2$

Step 1.5.2

Add $16$ and $2$ .

$u_{upper} = 18$

Step 1.6

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

$u_{lower} = 6$

$u_{upper} = 18$

Step 1.7

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

$\int_{6}^{18} u^{3} \frac{1}{2} d u$

Step 2

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

$\int_{6}^{18} \frac{u^{3}}{2} d u$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{6}^{18} u^{3} d u$

Step 4

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{2} \frac{1}{4} u^{4}]_{6}^{18}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{4} u^{4}$ at $18$ and at $6$ .

$\frac{1}{2} ((\frac{1}{4} \cdot 18^{4}) - \frac{1}{4} \cdot 6^{4})$

Step 5.2

Simplify.

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Step 5.2.1

Raise $18$ to the power of $4$ .

$\frac{1}{2} (\frac{1}{4} \cdot 104976 - \frac{1}{4} \cdot 6^{4})$

Step 5.2.2

Combine $\frac{1}{4}$ and $104976$ .

$\frac{1}{2} (\frac{104976}{4} - \frac{1}{4} \cdot 6^{4})$

Step 5.2.3

Cancel the common factor of $104976$ and $4$ .

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Step 5.2.3.1

Factor $4$ out of $104976$ .

$\frac{1}{2} (\frac{4 \cdot 26244}{4} - \frac{1}{4} \cdot 6^{4})$

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{2} (\frac{4 \cdot 26244}{4 (1)} - \frac{1}{4} \cdot 6^{4})$

Step 5.2.3.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{4 \cdot 26244}{4 \cdot 1} - \frac{1}{4} \cdot 6^{4})$

Step 5.2.3.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{26244}{1} - \frac{1}{4} \cdot 6^{4})$

Step 5.2.3.2.4

Divide $26244$ by $1$ .

$\frac{1}{2} (26244 - \frac{1}{4} \cdot 6^{4})$

Step 5.2.4

Raise $6$ to the power of $4$ .

$\frac{1}{2} (26244 - \frac{1}{4} \cdot 1296)$

Step 5.2.5

Multiply $1296$ by $- 1$ .

$\frac{1}{2} (26244 - 1296 (\frac{1}{4}))$

Step 5.2.6

Combine $- 1296$ and $\frac{1}{4}$ .

$\frac{1}{2} (26244 + \frac{- 1296}{4})$

Step 5.2.7

Cancel the common factor of $- 1296$ and $4$ .

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Step 5.2.7.1

Factor $4$ out of $- 1296$ .

$\frac{1}{2} (26244 + \frac{4 \cdot - 324}{4})$

Step 5.2.7.2

Cancel the common factors.

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Step 5.2.7.2.1

Factor $4$ out of $4$ .

$\frac{1}{2} (26244 + \frac{4 \cdot - 324}{4 (1)})$

Step 5.2.7.2.2

Cancel the common factor.

$\frac{1}{2} (26244 + \frac{4 \cdot - 324}{4 \cdot 1})$

Step 5.2.7.2.3

Rewrite the expression.

$\frac{1}{2} (26244 + \frac{- 324}{1})$

Step 5.2.7.2.4

Divide $- 324$ by $1$ .

$\frac{1}{2} (26244 - 324)$

Step 5.2.8

Subtract $324$ from $26244$ .

$\frac{1}{2} \cdot 25920$

Step 5.2.9

Combine $\frac{1}{2}$ and $25920$ .

$\frac{25920}{2}$

Step 5.2.10

Cancel the common factor of $25920$ and $2$ .

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Step 5.2.10.1

Factor $2$ out of $25920$ .

$\frac{2 \cdot 12960}{2}$

Step 5.2.10.2

Cancel the common factors.

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Step 5.2.10.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 12960}{2 (1)}$

Step 5.2.10.2.2

Cancel the common factor.

$\frac{2 \cdot 12960}{2 \cdot 1}$

Step 5.2.10.2.3

Rewrite the expression.

$\frac{12960}{1}$

Step 5.2.10.2.4

Divide $12960$ by $1$ .

$12960$

Step 6