Calculus Examples

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

Step 1

This integral could not be completed using integration by parts. Mathway will use another method.

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $2 x - 1$ .

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

Step 2.1.2

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 2.1.3

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

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Step 2.1.3.1

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 2.1.3.2

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 2.1.3.3

Multiply $2$ by $1$ .

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

Step 2.1.4

Differentiate using the Constant Rule.

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Step 2.1.4.1

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

$2 + 0$

Step 2.1.4.2

Add $2$ and $0$ .

$2$

Step 2.2

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

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

Step 3

Combine $\frac{1}{2}$ and $u^{7}$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $\frac{u}{2}$ by $\frac{u^{7}}{2}$ .

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

Step 4.3

Raise $u$ to the power of $1$ .

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

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.5

Add $1$ and $7$ .

$\int \frac{u^{8}}{2 \cdot 2} + \frac{1}{2} \cdot \frac{u^{7}}{2} d u$

Step 4.6

Multiply $2$ by $2$ .

$\int \frac{u^{8}}{4} + \frac{1}{2} \cdot \frac{u^{7}}{2} d u$

Step 4.7

Multiply $\frac{1}{2}$ by $\frac{u^{7}}{2}$ .

$\int \frac{u^{8}}{4} + \frac{u^{7}}{2 \cdot 2} d u$

Step 4.8

Multiply $2$ by $2$ .

$\int \frac{u^{8}}{4} + \frac{u^{7}}{4} d u$

Step 5

Split the single integral into multiple integrals.

$\int \frac{u^{8}}{4} d u + \int \frac{u^{7}}{4} d u$

Step 6

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int u^{8} d u + \int \frac{u^{7}}{4} d u$

Step 7

By the Power Rule, the integral of $u^{8}$ with respect to $u$ is $\frac{1}{9} u^{9}$ .

$\frac{1}{4} (\frac{1}{9} u^{9} + C) + \int \frac{u^{7}}{4} d u$

Step 8

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} (\frac{1}{9} u^{9} + C) + \frac{1}{4} \int u^{7} d u$

Step 9

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

$\frac{1}{4} (\frac{1}{9} u^{9} + C) + \frac{1}{4} (\frac{1}{8} u^{8} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{1}{4} \cdot \frac{1}{9} u^{9} + \frac{1}{4} \cdot \frac{1}{8} u^{8} + C$

Step 10.2

Simplify.

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Step 10.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{9}$ .

$\frac{1}{4 \cdot 9} u^{9} + \frac{1}{4} \cdot \frac{1}{8} u^{8} + C$

Step 10.2.2

Multiply $4$ by $9$ .

$\frac{1}{36} u^{9} + \frac{1}{4} \cdot \frac{1}{8} u^{8} + C$

Step 10.2.3

Multiply $\frac{1}{4}$ by $\frac{1}{8}$ .

$\frac{1}{36} u^{9} + \frac{1}{4 \cdot 8} u^{8} + C$

Step 10.2.4

Multiply $4$ by $8$ .

$\frac{1}{36} u^{9} + \frac{1}{32} u^{8} + C$

Step 11

Replace all occurrences of $u$ with $2 x - 1$ .

$\frac{1}{36} {(2 x - 1)}^{9} + \frac{1}{32} {(2 x - 1)}^{8} + C$