Calculus Examples

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

Step 1

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

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

Step 2

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

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

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Step 2.1.1

Differentiate $4 + x^{2}$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

$0 + \frac{d}{d x} [x^{2}]$

Step 2.1.4

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

$0 + 2 x$

Step 2.1.5

Add $0$ and $2 x$ .

$2 x$

Step 2.2

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

$u_{lower} = 4 + 1^{2}$

Step 2.3

Simplify.

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Step 2.3.1

One to any power is one.

$u_{lower} = 4 + 1$

Step 2.3.2

Add $4$ and $1$ .

$u_{lower} = 5$

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

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

$u_{upper} = 4 + 4$

Step 2.5.2

Add $4$ and $4$ .

$u_{upper} = 8$

Step 2.6

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

$u_{lower} = 5$

$u_{upper} = 8$

Step 2.7

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

$2 \int_{5}^{8} \sqrt{u} \frac{1}{2} d u$

Step 3

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

$2 \int_{5}^{8} \frac{\sqrt{u}}{2} d u$

Step 4

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

$2 (\frac{1}{2} \int_{5}^{8} \sqrt{u} d u)$

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 $2$ .

$\frac{2}{2} \int_{5}^{8} \sqrt{u} d u$

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Cancel the common factor.

$\frac{2}{2} \int_{5}^{8} \sqrt{u} d u$

Step 5.1.2.2

Rewrite the expression.

$1 \int_{5}^{8} \sqrt{u} d u$

Step 5.1.3

Multiply $\int_{5}^{8} \sqrt{u} d u$ by $1$ .

$\int_{5}^{8} \sqrt{u} d u$

Step 5.2

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

$\int_{5}^{8} u^{\frac{1}{2}} d u$

Step 6

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

$\frac{2}{3} u^{\frac{3}{2}}]_{5}^{8}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $8$ and at $5$ .

$(\frac{2}{3} \cdot 8^{\frac{3}{2}}) - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2

Simplify.

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Step 7.2.1

Combine $\frac{2}{3}$ and $8^{\frac{3}{2}}$ .

$\frac{2 \cdot 8^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.2

Rewrite $8$ as $2^{3}$ .

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

Step 7.2.3

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

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Step 7.2.3.1

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

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

Step 7.2.3.2

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

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Step 7.2.3.2.1

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

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

Step 7.2.3.2.2

Multiply $3$ by $3$ .

$\frac{2 \cdot 2^{\frac{9}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.4

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

$\frac{2^{1 + \frac{9}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.5

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

$\frac{2^{\frac{2}{2} + \frac{9}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.6

Combine the numerators over the common denominator.

$\frac{2^{\frac{2 + 9}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.7

Add $2$ and $9$ .

$\frac{2^{\frac{11}{2}}}{3} - \frac{2}{3} \cdot 5^{\frac{3}{2}}$

Step 7.2.8

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

$\frac{2^{\frac{11}{2}}}{3} - \frac{5^{\frac{3}{2}} \cdot 2}{3}$

Step 7.2.9

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

$\frac{2^{\frac{11}{2}}}{3} - \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{\frac{11}{2}}}{3} - \frac{2 \cdot 5^{\frac{3}{2}}}{3}$

Decimal Form:

$7.63138474 \dots$

Step 9