Calculus Examples

$\int (- 6 x + 2) {(- 3 x^{2} + 2 x + 5)}^{3} d x$

Step 1

Let $u = - 3 x^{2} + 2 x + 5$ . Then $d u = (- 6 x + 2) d x$ , so $\frac{1}{- 6 x + 2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = - 3 x^{2} + 2 x + 5$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $- 3 x^{2} + 2 x + 5$ .

$\frac{d}{d x} [- 3 x^{2} + 2 x + 5]$

Step 1.1.2

By the Sum Rule, the derivative of $- 3 x^{2} + 2 x + 5$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [5]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 1.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$- 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [5]$

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 $n = 2$ .

$- 3 (2 x) + \frac{d}{d x} [2 x] + \frac{d}{d x} [5]$

Step 1.1.3.3

Multiply $2$ by $- 3$ .

$- 6 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [5]$

Step 1.1.4

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

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Step 1.1.4.1

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

$- 6 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.1.4.2

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

$- 6 x + 2 \cdot 1 + \frac{d}{d x} [5]$

Step 1.1.4.3

Multiply $2$ by $1$ .

$- 6 x + 2 + \frac{d}{d x} [5]$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

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

$- 6 x + 2 + 0$

Step 1.1.5.2

Add $- 6 x + 2$ and $0$ .

$- 6 x + 2$

Step 1.2

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

$\int u^{3} d u$

Step 2

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

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

Step 3

Replace all occurrences of $u$ with $- 3 x^{2} + 2 x + 5$ .

$\frac{1}{4} {(- 3 x^{2} + 2 x + 5)}^{4} + C$