Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $2 x + 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

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 $n = 1$ .

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

Step 1.1.3.3

Multiply $2$ by $1$ .

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

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$2 + 0$

Step 1.1.4.2

Add $2$ and $0$ .

$2$

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Multiply $2$ by $0$ .

$u_{lower} = 0 + 1$

Step 1.3.2

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Multiply $2$ by $2$ .

$u_{upper} = 4 + 1$

Step 1.5.2

Add $4$ and $1$ .

$u_{upper} = 5$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 5$

Step 1.7

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

$\frac{1}{2} \frac{2}{3} u^{\frac{3}{2}}]_{1}^{5}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $5$ and at $1$ .

$\frac{1}{2} ((\frac{2}{3} \cdot 5^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

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

$\frac{1}{2} (\frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 6.2.2

One to any power is one.

$\frac{1}{2} (\frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3} \cdot 1)$

Step 6.2.3

Multiply $- 1$ by $1$ .

$\frac{1}{2} (\frac{2 \cdot 5^{\frac{3}{2}}}{3} - \frac{2}{3})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Combine.

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

Step 7.3

Cancel the common factor of $2$ .

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Step 7.3.1

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

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

Step 7.3.2

Factor $2$ out of $- 2$ .

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

Step 7.3.3

Cancel the common factor.

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

Step 7.3.4

Rewrite the expression.

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

Step 7.4

Simplify each term.

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Step 7.4.1

Cancel the common factor.

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

Step 7.4.2

Rewrite the expression.

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

Step 7.4.3

Multiply $5^{\frac{3}{2}}$ by $1$ .

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

Step 7.4.4

Move the negative in front of the fraction.

$\frac{5^{\frac{3}{2}}}{3} - \frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{5^{\frac{3}{2}}}{3} - \frac{1}{3}$

Decimal Form:

$3.39344662 \dots$

Step 9