Calculus Examples

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

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2

Let $u = x + 6$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x + 6$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x + 6$ .

$\frac{d}{d x} [x + 6]$

Step 2.1.2

By the Sum Rule, the derivative of $x + 6$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 2.1.3

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

$1 + \frac{d}{d x} [6]$

Step 2.1.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

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

$3 \int (u - 6) u^{2} d u$

Step 3

Expand $(u - 6) u^{2}$ .

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Step 3.1

Apply the distributive property.

$3 \int u \cdot u^{2} - 6 u^{2} d u$

Step 3.2

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

$3 \int u^{1} u^{2} - 6 u^{2} d u$

Step 3.3

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

$3 \int u^{1 + 2} - 6 u^{2} d u$

Step 3.4

Add $1$ and $2$ .

$3 \int u^{3} - 6 u^{2} d u$

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

$3 (\frac{1}{4} u^{4} + C + \int - 6 u^{2} d u)$

Step 6

Since $- 6$ is constant with respect to $u$ , move $- 6$ out of the integral.

$3 (\frac{1}{4} u^{4} + C - 6 \int u^{2} d u)$

Step 7

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$3 (\frac{1}{4} u^{4} + C - 6 (\frac{1}{3} u^{3} + C))$

Step 8

Simplify.

$3 (\frac{u^{4}}{4} - 2 u^{3}) + C$

Step 9

Replace all occurrences of $u$ with $x + 6$ .

$3 (\frac{{(x + 6)}^{4}}{4} - 2 {(x + 6)}^{3}) + C$

Step 10

Reorder terms.

$3 (\frac{1}{4} {(x + 6)}^{4} - 2 {(x + 6)}^{3}) + C$