Calculus Examples

$(x - 1) {(x^{2} - 2 x + 5)}^{5}$

Step 1

Let $u = x^{2} - 2 x + 5$ . Then $d u = (2 x - 2) d x$ , so $\frac{1}{2} d u = (x - 1) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{2} - 2 x + 5$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} - 2 x + 5$ .

$\frac{d}{d x} [x^{2} - 2 x + 5]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $x^{2} - 2 x + 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [5]$

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

$2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

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 $n = 1$ .

$2 x - 2 \cdot 1 + \frac{d}{d x} [5]$

Step 1.1.3.3

Multiply $- 2$ by $1$ .

$2 x - 2 + \frac{d}{d x} [5]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$2 x - 2 + 0$

Step 1.1.4.2

Add $2 x - 2$ and $0$ .

$2 x - 2$

Step 1.2

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

$\int u^{5} \frac{1}{2} d u$

Step 2

Combine $u^{5}$ and $\frac{1}{2}$ .

$\int \frac{u^{5}}{2} d u$

Step 3

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

$\frac{1}{2} \int u^{5} d u$

Step 4

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

$\frac{1}{2} (\frac{1}{6} u^{6} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{1}{2} (\frac{1}{6} u^{6} + C)$ as $\frac{1}{2} \cdot \frac{1}{6} u^{6} + C$ .

$\frac{1}{2} \cdot \frac{1}{6} u^{6} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{6}$ .

$\frac{1}{2 \cdot 6} u^{6} + C$

Step 5.2.2

Multiply $2$ by $6$ .

$\frac{1}{12} u^{6} + C$

Step 6

Replace all occurrences of $u$ with $x^{2} - 2 x + 5$ .

$\frac{1}{12} {(x^{2} - 2 x + 5)}^{6} + C$