Calculus Examples

$\int_{1}^{7} x \sqrt{4 x^{2} + 9} d x$

Step 1

Let $u = 4 x^{2} + 9$ . Then $d u = 8 x d x$ , so $\frac{1}{8} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 x^{2} + 9$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x^{2} + 9$ .

$\frac{d}{d x} [4 x^{2} + 9]$

Step 1.1.2

By the Sum Rule, the derivative of $4 x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [9]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$

Step 1.1.3.2

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

$4 (2 x) + \frac{d}{d x} [9]$

Step 1.1.3.3

Multiply $2$ by $4$ .

$8 x + \frac{d}{d x} [9]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$8 x + 0$

Step 1.1.4.2

Add $8 x$ and $0$ .

$8 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = 4 x^{2} + 9$ .

$u_{lower} = 4 \cdot 1^{2} + 9$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

One to any power is one.

$u_{lower} = 4 \cdot 1 + 9$

Step 1.3.1.2

Multiply $4$ by $1$ .

$u_{lower} = 4 + 9$

Step 1.3.2

Add $4$ and $9$ .

$u_{lower} = 13$

Step 1.4

Substitute the upper limit in for $x$ in $u = 4 x^{2} + 9$ .

$u_{upper} = 4 \cdot 7^{2} + 9$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Raise $7$ to the power of $2$ .

$u_{upper} = 4 \cdot 49 + 9$

Step 1.5.1.2

Multiply $4$ by $49$ .

$u_{upper} = 196 + 9$

Step 1.5.2

Add $196$ and $9$ .

$u_{upper} = 205$

Step 1.6

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

$u_{lower} = 13$

$u_{upper} = 205$

Step 1.7

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

$\int_{13}^{205} \sqrt{u} \frac{1}{8} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{8}$ .

$\int_{13}^{205} \frac{\sqrt{u}}{8} d u$

Step 3

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{8} \int_{13}^{205} \sqrt{u} d u$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\frac{1}{8} \int_{13}^{205} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{8} \frac{2}{3} u^{\frac{3}{2}}]_{13}^{205}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $205$ and at $13$ .

$\frac{1}{8} ((\frac{2}{3} \cdot 205^{\frac{3}{2}}) - \frac{2}{3} \cdot 13^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Combine $\frac{2}{3}$ and $205^{\frac{3}{2}}$ .

$\frac{1}{8} (\frac{2 \cdot 205^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 13^{\frac{3}{2}})$

Step 6.2.2

Combine $13^{\frac{3}{2}}$ and $\frac{2}{3}$ .

$\frac{1}{8} (\frac{2 \cdot 205^{\frac{3}{2}}}{3} - \frac{13^{\frac{3}{2}} \cdot 2}{3})$

Step 6.2.3

Move $2$ to the left of $13^{\frac{3}{2}}$ .

$\frac{1}{8} (\frac{2 \cdot 205^{\frac{3}{2}}}{3} - \frac{2 \cdot 13^{\frac{3}{2}}}{3})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1}{8} \cdot \frac{2 \cdot 205^{\frac{3}{2}}}{3} + \frac{1}{8} (- \frac{2 \cdot 13^{\frac{3}{2}}}{3})$

Step 7.2

Combine.

$\frac{1 (2 \cdot 205^{\frac{3}{2}})}{8 \cdot 3} + \frac{1}{8} (- \frac{2 \cdot 13^{\frac{3}{2}}}{3})$

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

Move the leading negative in $- \frac{2 \cdot 13^{\frac{3}{2}}}{3}$ into the numerator.

$\frac{1 (2 \cdot 205^{\frac{3}{2}})}{8 \cdot 3} + \frac{1}{8} \cdot \frac{- 2 \cdot 13^{\frac{3}{2}}}{3}$

Step 7.3.2

Factor $2$ out of $8$ .

$\frac{1 (2 \cdot 205^{\frac{3}{2}})}{8 \cdot 3} + \frac{1}{2 (4)} \cdot \frac{- 2 \cdot 13^{\frac{3}{2}}}{3}$

Step 7.3.3

Factor $2$ out of $- 2 \cdot 13^{\frac{3}{2}}$ .

$\frac{1 (2 \cdot 205^{\frac{3}{2}})}{8 \cdot 3} + \frac{1}{2 (4)} \cdot \frac{2 (- 13^{\frac{3}{2}})}{3}$

Step 7.3.4

Cancel the common factor.

$\frac{1 (2 \cdot 205^{\frac{3}{2}})}{8 \cdot 3} + \frac{1}{2 \cdot 4} \cdot \frac{2 (- 13^{\frac{3}{2}})}{3}$

Step 7.3.5

Rewrite the expression.

$\frac{1 (2 \cdot 205^{\frac{3}{2}})}{8 \cdot 3} + \frac{1}{4} \cdot \frac{- 13^{\frac{3}{2}}}{3}$

Step 7.4

Multiply $\frac{1}{4}$ by $\frac{- 13^{\frac{3}{2}}}{3}$ .

$\frac{1 (2 \cdot 205^{\frac{3}{2}})}{8 \cdot 3} + \frac{- 13^{\frac{3}{2}}}{4 \cdot 3}$

Step 7.5

Multiply $4$ by $3$ .

$\frac{1 (2 \cdot 205^{\frac{3}{2}})}{8 \cdot 3} + \frac{- 13^{\frac{3}{2}}}{12}$

Step 7.6

Simplify each term.

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Step 7.6.1

Factor $2$ out of $1 (2 \cdot 205^{\frac{3}{2}})$ .

$\frac{2 (1 \cdot (205^{\frac{3}{2}}))}{8 \cdot 3} + \frac{- 13^{\frac{3}{2}}}{12}$

Step 7.6.2

Cancel the common factors.

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Step 7.6.2.1

Factor $2$ out of $8 \cdot 3$ .

$\frac{2 (1 \cdot (205^{\frac{3}{2}}))}{2 (4 \cdot 3)} + \frac{- 13^{\frac{3}{2}}}{12}$

Step 7.6.2.2

Cancel the common factor.

$\frac{2 (1 \cdot (205^{\frac{3}{2}}))}{2 (4 \cdot 3)} + \frac{- 13^{\frac{3}{2}}}{12}$

Step 7.6.2.3

Rewrite the expression.

$\frac{1 \cdot (205^{\frac{3}{2}})}{4 \cdot 3} + \frac{- 13^{\frac{3}{2}}}{12}$

Step 7.6.3

Multiply $205^{\frac{3}{2}}$ by $1$ .

$\frac{205^{\frac{3}{2}}}{4 \cdot 3} + \frac{- 13^{\frac{3}{2}}}{12}$

Step 7.6.4

Multiply $4$ by $3$ .

$\frac{205^{\frac{3}{2}}}{12} + \frac{- 13^{\frac{3}{2}}}{12}$

Step 7.6.5

Move the negative in front of the fraction.

$\frac{205^{\frac{3}{2}}}{12} - \frac{13^{\frac{3}{2}}}{12}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{205^{\frac{3}{2}}}{12} - \frac{13^{\frac{3}{2}}}{12}$

Decimal Form:

$240.69009594 \dots$

Step 9