Calculus Examples

$\int_{0}^{7} \sqrt{7 x} d x$

Step 1

Let $u = 7 x$ . Then $d u = 7 d x$ , so $\frac{1}{7} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 7 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $7 x$ .

$\frac{d}{d x} [7 x]$

Step 1.1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$7 \frac{d}{d x} [x]$

Step 1.1.3

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

$7 \cdot 1$

Step 1.1.4

Multiply $7$ by $1$ .

$7$

Step 1.2

Substitute the lower limit in for $x$ in $u = 7 x$ .

$u_{lower} = 7 \cdot 0$

Step 1.3

Multiply $7$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = 7 x$ .

$u_{upper} = 7 \cdot 7$

Step 1.5

Multiply $7$ by $7$ .

$u_{upper} = 49$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 49$

Step 1.7

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

$\int_{0}^{49} \sqrt{u} \frac{1}{7} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{7}$ .

$\int_{0}^{49} \frac{\sqrt{u}}{7} d u$

Step 3

Since $\frac{1}{7}$ is constant with respect to $u$ , move $\frac{1}{7}$ out of the integral.

$\frac{1}{7} \int_{0}^{49} \sqrt{u} d u$

Step 4

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

$\frac{1}{7} \int_{0}^{49} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{7} \frac{2}{3} u^{\frac{3}{2}}]_{0}^{49}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $49$ and at $0$ .

$\frac{1}{7} ((\frac{2}{3} \cdot 49^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Rewrite $49$ as $7^{2}$ .

$\frac{1}{7} (\frac{2}{3} \cdot {(7^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.2

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

$\frac{1}{7} (\frac{2}{3} \cdot 7^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Cancel the common factor.

$\frac{1}{7} (\frac{2}{3} \cdot 7^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.3.2

Rewrite the expression.

$\frac{1}{7} (\frac{2}{3} \cdot 7^{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.4

Raise $7$ to the power of $3$ .

$\frac{1}{7} (\frac{2}{3} \cdot 343 - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.5

Combine $\frac{2}{3}$ and $343$ .

$\frac{1}{7} (\frac{2 \cdot 343}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.6

Multiply $2$ by $343$ .

$\frac{1}{7} (\frac{686}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}})$

Step 6.2.7

Rewrite $0$ as $0^{2}$ .

$\frac{1}{7} (\frac{686}{3} - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}})$

Step 6.2.8

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

$\frac{1}{7} (\frac{686}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 6.2.9

Cancel the common factor of $2$ .

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Step 6.2.9.1

Cancel the common factor.

$\frac{1}{7} (\frac{686}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})})$

Step 6.2.9.2

Rewrite the expression.

$\frac{1}{7} (\frac{686}{3} - \frac{2}{3} \cdot 0^{3})$

Step 6.2.10

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

$\frac{1}{7} (\frac{686}{3} - \frac{2}{3} \cdot 0)$

Step 6.2.11

Multiply $0$ by $- 1$ .

$\frac{1}{7} (\frac{686}{3} + 0 (\frac{2}{3}))$

Step 6.2.12

Multiply $0$ by $\frac{2}{3}$ .

$\frac{1}{7} (\frac{686}{3} + 0)$

Step 6.2.13

Add $\frac{686}{3}$ and $0$ .

$\frac{1}{7} \cdot \frac{686}{3}$

Step 6.2.14

Multiply $\frac{1}{7}$ by $\frac{686}{3}$ .

$\frac{686}{7 \cdot 3}$

Step 6.2.15

Multiply $7$ by $3$ .

$\frac{686}{21}$

Step 6.2.16

Cancel the common factor of $686$ and $21$ .

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Step 6.2.16.1

Factor $7$ out of $686$ .

$\frac{7 (98)}{21}$

Step 6.2.16.2

Cancel the common factors.

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Step 6.2.16.2.1

Factor $7$ out of $21$ .

$\frac{7 \cdot 98}{7 \cdot 3}$

Step 6.2.16.2.2

Cancel the common factor.

$\frac{7 \cdot 98}{7 \cdot 3}$

Step 6.2.16.2.3

Rewrite the expression.

$\frac{98}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{98}{3}$

Decimal Form:

$32.\overline{6}$

Mixed Number Form:

$32 \frac{2}{3}$

Step 8