Calculus Examples

${(\frac{1}{2} x - 1)}^{7}$

Step 1

Write ${(\frac{1}{2} x - 1)}^{7}$ as a function.

$f (x) = {(\frac{1}{2} x - 1)}^{7}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int {(\frac{1}{2} x - 1)}^{7} d x$

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $\frac{1}{2} x - 1$ .

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

Step 4.1.2

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

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

Step 4.1.3

Evaluate $\frac{d}{d x} [\frac{1}{2} x]$ .

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Step 4.1.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} x$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

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

$\frac{1}{2} + 0$

Step 4.1.4.2

Add $\frac{1}{2}$ and $0$ .

$\frac{1}{2}$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$\int u^{7} \frac{1}{\frac{1}{2}} d u$

Step 5

Simplify.

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Step 5.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\int u^{7} (1 \cdot 2) d u$

Step 5.2

Multiply $2$ by $1$ .

$\int u^{7} \cdot 2 d u$

Step 5.3

Move $2$ to the left of $u^{7}$ .

$\int 2 u^{7} d u$

Step 6

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int u^{7} d u$

Step 7

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$2 (\frac{1}{8} u^{8} + C)$

Step 8

Simplify.

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Step 8.1

Rewrite $2 (\frac{1}{8} u^{8} + C)$ as $2 (\frac{1}{8}) u^{8} + C$ .

$2 (\frac{1}{8}) u^{8} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $2$ and $\frac{1}{8}$ .

$\frac{2}{8} u^{8} + C$

Step 8.2.2

Cancel the common factor of $2$ and $8$ .

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Step 8.2.2.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{8} u^{8} + C$

Step 8.2.2.2

Cancel the common factors.

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Step 8.2.2.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 1}{2 \cdot 4} u^{8} + C$

Step 8.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 4} u^{8} + C$

Step 8.2.2.2.3

Rewrite the expression.

$\frac{1}{4} u^{8} + C$

Step 9

Replace all occurrences of $u$ with $\frac{1}{2} x - 1$ .

$\frac{1}{4} {(\frac{1}{2} x - 1)}^{8} + C$

Step 10

The answer is the antiderivative of the function $f (x) = {(\frac{1}{2} x - 1)}^{7}$ .

$F (x) =$ $\frac{1}{4} {(\frac{1}{2} x - 1)}^{8} + C$