Calculus Examples

$\int_{3}^{6} 2 x \sqrt{x^{2} - 4} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{3}^{6} x \sqrt{x^{2} - 4} d x$

Step 2

Let $u = x^{2} - 4$ . 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 2.1

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

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Step 2.1.1

Differentiate $x^{2} - 4$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$2 x + 0$

Step 2.1.5

Add $2 x$ and $0$ .

$2 x$

Step 2.2

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

$u_{lower} = 3^{2} - 4$

Step 2.3

Simplify.

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Step 2.3.1

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

$u_{lower} = 9 - 4$

Step 2.3.2

Subtract $4$ from $9$ .

$u_{lower} = 5$

Step 2.4

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

$u_{upper} = 6^{2} - 4$

Step 2.5

Simplify.

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Step 2.5.1

Raise $6$ to the power of $2$ .

$u_{upper} = 36 - 4$

Step 2.5.2

Subtract $4$ from $36$ .

$u_{upper} = 32$

Step 2.6

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

$u_{lower} = 5$

$u_{upper} = 32$

Step 2.7

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

$2 \int_{5}^{32} \sqrt{u} \frac{1}{2} d u$

Step 3

Combine $\sqrt{u}$ and $\frac{1}{2}$ .

$2 \int_{5}^{32} \frac{\sqrt{u}}{2} d u$

Step 4

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

$2 (\frac{1}{2} \int_{5}^{32} \sqrt{u} d u)$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

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

$\frac{2}{2} \int_{5}^{32} \sqrt{u} d u$

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Cancel the common factor.

$\frac{2}{2} \int_{5}^{32} \sqrt{u} d u$

Step 5.1.2.2

Rewrite the expression.

$1 \int_{5}^{32} \sqrt{u} d u$

Step 5.1.3

Multiply $\int_{5}^{32} \sqrt{u} d u$ by $1$ .

$\int_{5}^{32} \sqrt{u} d u$

Step 5.2

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

$\int_{5}^{32} u^{\frac{1}{2}} d u$

Step 6

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

$\frac{2}{3} u^{\frac{3}{2}}]_{5}^{32}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $32$ and at $5$ .

$(\frac{2}{3} \cdot 32^{\frac{3}{2}}) - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2

Simplify.

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Step 7.2.1

Combine $\frac{2}{3}$ and $32^{\frac{3}{2}}$ .

$\frac{2 \cdot 32^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.2

Rewrite $32$ as $2^{5}$ .

$\frac{2 \cdot {(2^{5})}^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.3

Multiply the exponents in ${(2^{5})}^{\frac{3}{2}}$ .

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Step 7.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \cdot 2^{5 (\frac{3}{2})}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.3.2

Multiply $5 (\frac{3}{2})$ .

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Step 7.2.3.2.1

Combine $5$ and $\frac{3}{2}$ .

$\frac{2 \cdot 2^{\frac{5 \cdot 3}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.3.2.2

Multiply $5$ by $3$ .

$\frac{2 \cdot 2^{\frac{15}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2^{1 + \frac{15}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.5

Write $1$ as a fraction with a common denominator.

$\frac{2^{\frac{2}{2} + \frac{15}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.6

Combine the numerators over the common denominator.

$\frac{2^{\frac{2 + 15}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.7

Add $2$ and $15$ .

$\frac{2^{\frac{17}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.8

Combine $5^{\frac{3}{2}}$ and $\frac{2}{3}$ .

$\frac{2^{\frac{17}{2}}}{3} - \frac{5^{\frac{3}{2}} \cdot 2}{3}$

Step 7.2.9

Move $2$ to the left of $5^{\frac{3}{2}}$ .

$\frac{2^{\frac{17}{2}}}{3} - \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{\frac{17}{2}}}{3} - \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Decimal Form:

$113.22599739 \dots$

Step 9