Calculus Examples

$\int x {(x - 1)}^{5} d x$

Step 1

Let $u = x - 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $x - 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

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

$\int (u + 1) u^{5} d u$

Step 2

Multiply $(u + 1) u^{5}$ .

$\int u \cdot u^{5} + 1 u^{5} d u$

Step 3

Simplify.

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Step 3.1

Multiply $u$ by $u^{5}$ by adding the exponents.

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Step 3.1.1

Multiply $u$ by $u^{5}$ .

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Step 3.1.1.1

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

$\int u^{1} u^{5} + 1 u^{5} d u$

Step 3.1.1.2

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

$\int u^{1 + 5} + 1 u^{5} d u$

Step 3.1.2

Add $1$ and $5$ .

$\int u^{6} + 1 u^{5} d u$

Step 3.2

Multiply $u^{5}$ by $1$ .

$\int u^{6} + u^{5} d u$

Step 4

Split the single integral into multiple integrals.

$\int u^{6} d u + \int u^{5} d u$

Step 5

By the Power Rule, the integral of $u^{6}$ with respect to $u$ is $\frac{1}{7} u^{7}$ .

$\frac{1}{7} u^{7} + C + \int u^{5} d u$

Step 6

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$\frac{1}{7} u^{7} + C + \frac{1}{6} u^{6} + C$

Step 7

Simplify.

$\frac{1}{7} u^{7} + \frac{1}{6} u^{6} + C$

Step 8

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

$\frac{1}{7} {(x - 1)}^{7} + \frac{1}{6} {(x - 1)}^{6} + C$