Calculus Examples

$\int_{0}^{7} \sqrt{4 + 3 x} d x$

Step 1

Let $u = 4 + 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 + 3 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 + 3 x$ .

$\frac{d}{d x} [4 + 3 x]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $4 + 3 x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [3 x]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [3 x]$

Step 1.1.2.2

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

$0 + \frac{d}{d x} [3 x]$

Step 1.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$0 + 3 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 + 3 \cdot 1$

Step 1.1.3.3

Multiply $3$ by $1$ .

$0 + 3$

Step 1.1.4

Add $0$ and $3$ .

$3$

Step 1.2

Substitute the lower limit in for $x$ in $u = 4 + 3 x$ .

$u_{lower} = 4 + 3 \cdot 0$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $3$ by $0$ .

$u_{lower} = 4 + 0$

Step 1.3.2

Add $4$ and $0$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $x$ in $u = 4 + 3 x$ .

$u_{upper} = 4 + 3 \cdot 7$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $3$ by $7$ .

$u_{upper} = 4 + 21$

Step 1.5.2

Add $4$ and $21$ .

$u_{upper} = 25$

Step 1.6

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

$u_{lower} = 4$

$u_{upper} = 25$

Step 1.7

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

$\int_{4}^{25} \sqrt{u} \frac{1}{3} d u$

Step 2

Combine $\sqrt{u}$ and $\frac{1}{3}$ .

$\int_{4}^{25} \frac{\sqrt{u}}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{4}^{25} \sqrt{u} d u$

Step 4

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

$\frac{1}{3} \int_{4}^{25} u^{\frac{1}{2}} d u$

Step 5

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

$\frac{1}{3} \frac{2}{3} u^{\frac{3}{2}}]_{4}^{25}$

Step 6

Simplify the expression.

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Step 6.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $25$ and at $4$ .

$\frac{1}{3} ((\frac{2}{3} \cdot 25^{\frac{3}{2}}) - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Rewrite $25$ as $5^{2}$ .

$\frac{1}{3} (\frac{2}{3} \cdot {(5^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 6.2.2

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

$\frac{1}{3} (\frac{2}{3} \cdot 5^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 6.2.3

Cancel the common factor of $2$ .

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Step 6.2.3.1

Cancel the common factor.

$\frac{1}{3} (\frac{2}{3} \cdot 5^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 6.2.3.2

Rewrite the expression.

$\frac{1}{3} (\frac{2}{3} \cdot 5^{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 6.2.4

Raise $5$ to the power of $3$ .

$\frac{1}{3} (\frac{2}{3} \cdot 125 - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 6.3

Simplify.

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Step 6.3.1

Combine $\frac{2}{3}$ and $125$ .

$\frac{1}{3} (\frac{2 \cdot 125}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 6.3.2

Multiply $2$ by $125$ .

$\frac{1}{3} (\frac{250}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}})$

Step 6.4

Simplify.

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Step 6.4.1

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

$\frac{1}{3} (\frac{250}{3} - \frac{2}{3} \cdot {(2^{2})}^{\frac{3}{2}})$

Step 6.4.2

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

$\frac{1}{3} (\frac{250}{3} - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})})$

Step 6.4.3

Cancel the common factor of $2$ .

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Step 6.4.3.1

Cancel the common factor.

$\frac{1}{3} (\frac{250}{3} - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})})$

Step 6.4.3.2

Rewrite the expression.

$\frac{1}{3} (\frac{250}{3} - \frac{2}{3} \cdot 2^{3})$

Step 6.4.4

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

$\frac{1}{3} (\frac{250}{3} - \frac{2}{3} \cdot 8)$

Step 6.5

Simplify.

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Step 6.5.1

Multiply $8$ by $- 1$ .

$\frac{1}{3} (\frac{250}{3} - 8 (\frac{2}{3}))$

Step 6.5.2

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

$\frac{1}{3} (\frac{250}{3} + \frac{- 8 \cdot 2}{3})$

Step 6.5.3

Multiply $- 8$ by $2$ .

$\frac{1}{3} (\frac{250}{3} + \frac{- 16}{3})$

Step 6.5.4

Move the negative in front of the fraction.

$\frac{1}{3} (\frac{250}{3} - \frac{16}{3})$

Step 6.6

Simplify.

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Step 6.6.1

Combine the numerators over the common denominator.

$\frac{1}{3} \cdot \frac{250 - 16}{3}$

Step 6.6.2

Subtract $16$ from $250$ .

$\frac{1}{3} \cdot \frac{234}{3}$

Step 6.6.3

Cancel the common factor of $234$ and $3$ .

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Step 6.6.3.1

Factor $3$ out of $234$ .

$\frac{1}{3} \cdot \frac{3 \cdot 78}{3}$

Step 6.6.3.2

Cancel the common factors.

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Step 6.6.3.2.1

Factor $3$ out of $3$ .

$\frac{1}{3} \cdot \frac{3 \cdot 78}{3 (1)}$

Step 6.6.3.2.2

Cancel the common factor.

$\frac{1}{3} \cdot \frac{3 \cdot 78}{3 \cdot 1}$

Step 6.6.3.2.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{78}{1}$

Step 6.6.3.2.4

Divide $78$ by $1$ .

$\frac{1}{3} \cdot 78$

Step 6.7

Simplify.

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Step 6.7.1

Combine $\frac{1}{3}$ and $78$ .

$\frac{78}{3}$

Step 6.7.2

Cancel the common factor of $78$ and $3$ .

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Step 6.7.2.1

Factor $3$ out of $78$ .

$\frac{3 \cdot 26}{3}$

Step 6.7.2.2

Cancel the common factors.

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Step 6.7.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 26}{3 (1)}$

Step 6.7.2.2.2

Cancel the common factor.

$\frac{3 \cdot 26}{3 \cdot 1}$

Step 6.7.2.2.3

Rewrite the expression.

$\frac{26}{1}$

Step 6.7.2.2.4

Divide $26$ by $1$ .

$26$