Calculus Examples

$\int_{0}^{3} \frac{3}{2} \sqrt{x + 1} d x$

Step 1

Since $\frac{3}{2}$ is constant with respect to $x$ , move $\frac{3}{2}$ out of the integral.

$\frac{3}{2} \int_{0}^{3} \sqrt{x + 1} d x$

Step 2

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $x + 1$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

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

$u_{lower} = 0 + 1$

Step 2.3

Add $0$ and $1$ .

$u_{lower} = 1$

Step 2.4

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

$u_{upper} = 3 + 1$

Step 2.5

Add $3$ and $1$ .

$u_{upper} = 4$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = 4$

Step 2.7

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

$\frac{3}{2} \int_{1}^{4} \sqrt{u} d u$

Step 3

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

$\frac{3}{2} \int_{1}^{4} u^{\frac{1}{2}} d u$

Step 4

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

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

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $4$ and at $1$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Rewrite $4$ as $2^{2}$ .

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

Step 5.2.2

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

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

Step 5.2.3

Cancel the common factor of $2$ .

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Step 5.2.3.1

Cancel the common factor.

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

Step 5.2.3.2

Rewrite the expression.

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

Step 5.2.4

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

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

Step 5.2.5

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

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

Step 5.2.6

Multiply $2$ by $8$ .

$\frac{3}{2} (\frac{16}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.7

One to any power is one.

$\frac{3}{2} (\frac{16}{3} - \frac{2}{3} \cdot 1)$

Step 5.2.8

Multiply $- 1$ by $1$ .

$\frac{3}{2} (\frac{16}{3} - \frac{2}{3})$

Step 5.2.9

Combine the numerators over the common denominator.

$\frac{3}{2} \cdot \frac{16 - 2}{3}$

Step 5.2.10

Subtract $2$ from $16$ .

$\frac{3}{2} \cdot \frac{14}{3}$

Step 5.2.11

Multiply $\frac{3}{2}$ by $\frac{14}{3}$ .

$\frac{3 \cdot 14}{2 \cdot 3}$

Step 5.2.12

Multiply $3$ by $14$ .

$\frac{42}{2 \cdot 3}$

Step 5.2.13

Multiply $2$ by $3$ .

$\frac{42}{6}$

Step 5.2.14

Cancel the common factor of $42$ and $6$ .

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Step 5.2.14.1

Factor $6$ out of $42$ .

$\frac{6 \cdot 7}{6}$

Step 5.2.14.2

Cancel the common factors.

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Step 5.2.14.2.1

Factor $6$ out of $6$ .

$\frac{6 \cdot 7}{6 (1)}$

Step 5.2.14.2.2

Cancel the common factor.

$\frac{6 \cdot 7}{6 \cdot 1}$

Step 5.2.14.2.3

Rewrite the expression.

$\frac{7}{1}$

Step 5.2.14.2.4

Divide $7$ by $1$ .

$7$

Step 6