Calculus Examples

$\int_{0}^{2} x \sqrt{1 + 2 x^{2}} d x$

Step 1

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

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Step 1.1

Let $u = 1 + 2 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 + 2 x^{2}$ .

$\frac{d}{d x} [1 + 2 x^{2}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 + 2 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [2 x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [2 x^{2}]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

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 $n = 2$ .

$0 + 2 (2 x)$

Step 1.1.3.3

Multiply $2$ by $2$ .

$0 + 4 x$

Step 1.1.4

Add $0$ and $4 x$ .

$4 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 + 2 x^{2}$ .

$u_{lower} = 1 + 2 \cdot 0^{2}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

$u_{lower} = 1 + 2 \cdot 0$

Step 1.3.1.2

Multiply $2$ by $0$ .

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 + 2 x^{2}$ .

$u_{upper} = 1 + 2 \cdot 2^{2}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Multiply $2$ by $2^{2}$ by adding the exponents.

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Step 1.5.1.1.1

Multiply $2$ by $2^{2}$ .

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Step 1.5.1.1.1.1

Raise $2$ to the power of $1$ .

$u_{upper} = 1 + 2^{1} \cdot 2^{2}$

Step 1.5.1.1.1.2

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

$u_{upper} = 1 + 2^{1 + 2}$

Step 1.5.1.1.2

Add $1$ and $2$ .

$u_{upper} = 1 + 2^{3}$

Step 1.5.1.2

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

$u_{upper} = 1 + 8$

Step 1.5.2

Add $1$ and $8$ .

$u_{upper} = 9$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 9$

Step 1.7

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

$\int_{1}^{9} \sqrt{u} \frac{1}{4} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{4}$ .

$\int_{1}^{9} \frac{\sqrt{u}}{4} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{1}^{9} \sqrt{u} d u$

Step 4

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

$\frac{1}{4} \int_{1}^{9} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{4} \frac{2}{3} u^{\frac{3}{2}}]_{1}^{9}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $9$ and at $1$ .

$\frac{1}{4} ((\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{1}{4} (\frac{2}{3} \cdot {(3^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.2

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

$\frac{1}{4} (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Cancel the common factor.

$\frac{1}{4} (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.3.2

Rewrite the expression.

$\frac{1}{4} (\frac{2}{3} \cdot 3^{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.4

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

$\frac{1}{4} (\frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.5

Combine $\frac{2}{3}$ and $27$ .

$\frac{1}{4} (\frac{2 \cdot 27}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.6

Multiply $2$ by $27$ .

$\frac{1}{4} (\frac{54}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.7

Cancel the common factor of $54$ and $3$ .

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Step 6.2.7.1

Factor $3$ out of $54$ .

$\frac{1}{4} (\frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.7.2

Cancel the common factors.

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Step 6.2.7.2.1

Factor $3$ out of $3$ .

$\frac{1}{4} (\frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.7.2.2

Cancel the common factor.

$\frac{1}{4} (\frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.7.2.3

Rewrite the expression.

$\frac{1}{4} (\frac{18}{1} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.7.2.4

Divide $18$ by $1$ .

$\frac{1}{4} (18 - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.8

One to any power is one.

$\frac{1}{4} (18 - \frac{2}{3} \cdot 1)$

Step 6.2.9

Multiply $- 1$ by $1$ .

$\frac{1}{4} (18 - \frac{2}{3})$

Step 6.2.10

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{4} (18 \cdot \frac{3}{3} - \frac{2}{3})$

Step 6.2.11

Combine $18$ and $\frac{3}{3}$ .

$\frac{1}{4} (\frac{18 \cdot 3}{3} - \frac{2}{3})$

Step 6.2.12

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{18 \cdot 3 - 2}{3}$

Step 6.2.13

Simplify the numerator.

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Step 6.2.13.1

Multiply $18$ by $3$ .

$\frac{1}{4} \cdot \frac{54 - 2}{3}$

Step 6.2.13.2

Subtract $2$ from $54$ .

$\frac{1}{4} \cdot \frac{52}{3}$

Step 6.2.14

Multiply $\frac{1}{4}$ by $\frac{52}{3}$ .

$\frac{52}{4 \cdot 3}$

Step 6.2.15

Multiply $4$ by $3$ .

$\frac{52}{12}$

Step 6.2.16

Cancel the common factor of $52$ and $12$ .

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Step 6.2.16.1

Factor $4$ out of $52$ .

$\frac{4 (13)}{12}$

Step 6.2.16.2

Cancel the common factors.

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Step 6.2.16.2.1

Factor $4$ out of $12$ .

$\frac{4 \cdot 13}{4 \cdot 3}$

Step 6.2.16.2.2

Cancel the common factor.

$\frac{4 \cdot 13}{4 \cdot 3}$

Step 6.2.16.2.3

Rewrite the expression.

$\frac{13}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{3}$

Decimal Form:

$4.\overline{3}$

Mixed Number Form:

$4 \frac{1}{3}$

Step 8