Calculus Examples

$\int_{1}^{7} {(2 x - 1)}^{4} d x$

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 $u = 2 x - 1$ .

$u_{lower} = 2 \cdot 1 - 1$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $2$ by $1$ .

$u_{lower} = 2 - 1$

Step 1.3.2

Subtract $1$ from $2$ .

$u_{lower} = 1$

Step 1.4

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

$u_{upper} = 2 \cdot 7 - 1$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $2$ by $7$ .

$u_{upper} = 14 - 1$

Step 1.5.2

Subtract $1$ from $14$ .

$u_{upper} = 13$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 13$

Step 1.7

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

$\int_{1}^{13} u^{4} \frac{1}{2} d u$

Step 2

Combine $u^{4}$ and $\frac{1}{2}$ .

$\int_{1}^{13} \frac{u^{4}}{2} d u$

Step 3

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

$\frac{1}{2} \int_{1}^{13} u^{4} d u$

Step 4

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{2} \frac{1}{5} u^{5}]_{1}^{13}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{5} u^{5}$ at $13$ and at $1$ .

$\frac{1}{2} ((\frac{1}{5} \cdot 13^{5}) - \frac{1}{5} \cdot 1^{5})$

Step 5.2

Simplify.

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Step 5.2.1

Raise $13$ to the power of $5$ .

$\frac{1}{2} (\frac{1}{5} \cdot 371293 - \frac{1}{5} \cdot 1^{5})$

Step 5.2.2

Combine $\frac{1}{5}$ and $371293$ .

$\frac{1}{2} (\frac{371293}{5} - \frac{1}{5} \cdot 1^{5})$

Step 5.2.3

One to any power is one.

$\frac{1}{2} (\frac{371293}{5} - \frac{1}{5} \cdot 1)$

Step 5.2.4

Multiply $- 1$ by $1$ .

$\frac{1}{2} (\frac{371293}{5} - \frac{1}{5})$

Step 5.2.5

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{371293 - 1}{5}$

Step 5.2.6

Subtract $1$ from $371293$ .

$\frac{1}{2} \cdot \frac{371292}{5}$

Step 5.2.7

Multiply $\frac{1}{2}$ by $\frac{371292}{5}$ .

$\frac{371292}{2 \cdot 5}$

Step 5.2.8

Multiply $2$ by $5$ .

$\frac{371292}{10}$

Step 5.2.9

Cancel the common factor of $371292$ and $10$ .

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Step 5.2.9.1

Factor $2$ out of $371292$ .

$\frac{2 (185646)}{10}$

Step 5.2.9.2

Cancel the common factors.

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Step 5.2.9.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 185646}{2 \cdot 5}$

Step 5.2.9.2.2

Cancel the common factor.

$\frac{2 \cdot 185646}{2 \cdot 5}$

Step 5.2.9.2.3

Rewrite the expression.

$\frac{185646}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{185646}{5}$

Decimal Form:

$37129.2$

Mixed Number Form:

$37129 \frac{1}{5}$

Step 7