Calculus Examples

${(6 x + 5)}^{2} d x$

Step 1

Write ${(6 x + 5)}^{2} d x$ as a function.

$f (x) = {(6 x + 5)}^{2} d x$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int {(6 x + 5)}^{2} d x d x$

Step 4

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

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

Step 5

Let $u = 6 x + 5$ . Then $d u = 6 d x$ , so $\frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 6 x + 5$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $6 x + 5$ .

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

Step 5.1.2

By the Sum Rule, the derivative of $6 x + 5$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [5]$ .

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

Step 5.1.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 5.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

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

Step 5.1.3.2

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

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

Step 5.1.3.3

Multiply $6$ by $1$ .

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

Step 5.1.4

Differentiate using the Constant Rule.

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Step 5.1.4.1

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

$6 + 0$

Step 5.1.4.2

Add $6$ and $0$ .

$6$

Step 5.2

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

$d \int u^{2} (\frac{u}{6} - \frac{5}{6}) \frac{1}{6} d u$

Step 6

Combine $\frac{1}{6}$ and $u^{2}$ .

$d \int \frac{u^{2}}{6} (\frac{u}{6} - \frac{5}{6}) d u$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$d \int \frac{u^{2}}{6} \cdot \frac{u}{6} + \frac{u^{2}}{6} (- \frac{5}{6}) d u$

Step 7.2

Reorder $\frac{u^{2}}{6}$ and $- 1$ .

$d \int \frac{u^{2}}{6} \cdot \frac{u}{6} - 1 \cdot \frac{u^{2}}{6} \frac{5}{6} d u$

Step 7.3

Multiply $\frac{u^{2}}{6}$ by $\frac{u}{6}$ .

$d \int \frac{u^{2} u}{6 \cdot 6} - 1 \frac{u^{2}}{6} \frac{5}{6} d u$

Step 7.4

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

$d \int \frac{u^{2} u^{1}}{6 \cdot 6} - 1 \frac{u^{2}}{6} \frac{5}{6} d u$

Step 7.5

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

$d \int \frac{u^{2 + 1}}{6 \cdot 6} - 1 \frac{u^{2}}{6} \frac{5}{6} d u$

Step 7.6

Add $2$ and $1$ .

$d \int \frac{u^{3}}{6 \cdot 6} - 1 \frac{u^{2}}{6} \frac{5}{6} d u$

Step 7.7

Multiply $6$ by $6$ .

$d \int \frac{u^{3}}{36} - 1 \frac{u^{2}}{6} \frac{5}{6} d u$

Step 7.8

Combine $- 1 \frac{u^{2}}{6}$ and $\frac{5}{6}$ .

$d \int \frac{u^{3}}{36} + \frac{- 1 \frac{u^{2}}{6} \cdot 5}{6} d u$

Step 7.9

Combine $\frac{u^{2}}{6}$ and $5$ .

$d \int \frac{u^{3}}{36} + \frac{- 1 \frac{u^{2} \cdot 5}{6}}{6} d u$

Step 8

Simplify.

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Step 8.1

Move $5$ to the left of $u^{2}$ .

$d \int \frac{u^{3}}{36} + \frac{- 1 \frac{5 \cdot u^{2}}{6}}{6} d u$

Step 8.2

Rewrite $- 1 \frac{5 u^{2}}{6}$ as $- \frac{5 u^{2}}{6}$ .

$d \int \frac{u^{3}}{36} + \frac{- \frac{5 u^{2}}{6}}{6} d u$

Step 8.3

Rewrite $\frac{- \frac{5 u^{2}}{6}}{6}$ as a product.

$d \int \frac{u^{3}}{36} - \frac{5 u^{2}}{6} \cdot \frac{1}{6} d u$

Step 8.4

Multiply $\frac{1}{6}$ by $\frac{5 u^{2}}{6}$ .

$d \int \frac{u^{3}}{36} - \frac{5 u^{2}}{6 \cdot 6} d u$

Step 8.5

Multiply $6$ by $6$ .

$d \int \frac{u^{3}}{36} - \frac{5 u^{2}}{36} d u$

Step 9

Split the single integral into multiple integrals.

$d (\int \frac{u^{3}}{36} d u + \int - \frac{5 u^{2}}{36} d u)$

Step 10

Since $\frac{1}{36}$ is constant with respect to $u$ , move $\frac{1}{36}$ out of the integral.

$d (\frac{1}{36} \int u^{3} d u + \int - \frac{5 u^{2}}{36} d u)$

Step 11

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

$d (\frac{1}{36} (\frac{1}{4} u^{4} + C) + \int - \frac{5 u^{2}}{36} d u)$

Step 12

Combine $\frac{1}{4}$ and $u^{4}$ .

$d (\frac{1}{36} (\frac{u^{4}}{4} + C) + \int - \frac{5 u^{2}}{36} d u)$

Step 13

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$d (\frac{1}{36} (\frac{u^{4}}{4} + C) - \int \frac{5 u^{2}}{36} d u)$

Step 14

Since $\frac{5}{36}$ is constant with respect to $u$ , move $\frac{5}{36}$ out of the integral.

$d (\frac{1}{36} (\frac{u^{4}}{4} + C) - (\frac{5}{36} \int u^{2} d u))$

Step 15

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

$d (\frac{1}{36} (\frac{u^{4}}{4} + C) - \frac{5}{36} (\frac{1}{3} u^{3} + C))$

Step 16

Simplify.

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Step 16.1

Combine $\frac{1}{3}$ and $u^{3}$ .

$d (\frac{1}{36} (\frac{u^{4}}{4} + C) - \frac{5}{36} (\frac{u^{3}}{3} + C))$

Step 16.2

Simplify.

$d (\frac{u^{4}}{144} - \frac{5 u^{3}}{108}) + C$

Step 17

Replace all occurrences of $u$ with $6 x + 5$ .

$d (\frac{{(6 x + 5)}^{4}}{144} - \frac{5 {(6 x + 5)}^{3}}{108}) + C$

Step 18

Reorder terms.

$d (\frac{1}{144} {(6 x + 5)}^{4} - \frac{5}{108} {(6 x + 5)}^{3}) + C$

Step 19

The answer is the antiderivative of the function $f (x) = {(6 x + 5)}^{2} d x$ .

$F (x) =$ $d (\frac{1}{144} {(6 x + 5)}^{4} - \frac{5}{108} {(6 x + 5)}^{3}) + C$