Calculus Examples

$\int x \sqrt[3]{6 x + 7} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sqrt[3]{6 x + 7}$ .

$x (\frac{1}{8} {(6 x + 7)}^{\frac{4}{3}}) - \int \frac{1}{8} {(6 x + 7)}^{\frac{4}{3}} d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{1}{8}$ and ${(6 x + 7)}^{\frac{4}{3}}$ .

$x \frac{{(6 x + 7)}^{\frac{4}{3}}}{8} - \int \frac{1}{8} {(6 x + 7)}^{\frac{4}{3}} d x$

Step 2.2

Combine $x$ and $\frac{{(6 x + 7)}^{\frac{4}{3}}}{8}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \int \frac{1}{8} {(6 x + 7)}^{\frac{4}{3}} d x$

Step 3

Since $\frac{1}{8}$ is constant with respect to $x$ , move $\frac{1}{8}$ out of the integral.

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - (\frac{1}{8} \int {(6 x + 7)}^{\frac{4}{3}} d x)$

Step 4

Let $u = 6 x + 7$ . 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 4.1

Let $u = 6 x + 7$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $6 x + 7$ .

$\frac{d}{d x} [6 x + 7]$

Step 4.1.2

By the Sum Rule, the derivative of $6 x + 7$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [7]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [7]$

Step 4.1.3

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

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Step 4.1.3.1

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

$6 \frac{d}{d x} [x] + \frac{d}{d x} [7]$

Step 4.1.3.2

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

$6 \cdot 1 + \frac{d}{d x} [7]$

Step 4.1.3.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [7]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$6 + 0$

Step 4.1.4.2

Add $6$ and $0$ .

$6$

Step 4.2

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

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{8} \int u^{\frac{4}{3}} \frac{1}{6} d u$

Step 5

Combine $u^{\frac{4}{3}}$ and $\frac{1}{6}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{8} \int \frac{u^{\frac{4}{3}}}{6} d u$

Step 6

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{8} (\frac{1}{6} \int u^{\frac{4}{3}} d u)$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{6}$ by $\frac{1}{8}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{6 \cdot 8} \int u^{\frac{4}{3}} d u$

Step 7.2

Multiply $6$ by $8$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{48} \int u^{\frac{4}{3}} d u$

Step 8

By the Power Rule, the integral of $u^{\frac{4}{3}}$ with respect to $u$ is $\frac{3}{7} u^{\frac{7}{3}}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{48} (\frac{3}{7} u^{\frac{7}{3}} + C)$

Step 9

Simplify.

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Step 9.1

Rewrite $\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{48} (\frac{3}{7} u^{\frac{7}{3}} + C)$ as $\frac{1}{8} x {(6 x + 7)}^{\frac{4}{3}} - \frac{1}{48} \cdot \frac{3}{7} u^{\frac{7}{3}} + C$ .

$\frac{1}{8} x {(6 x + 7)}^{\frac{4}{3}} - \frac{1}{48} \cdot \frac{3}{7} u^{\frac{7}{3}} + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $\frac{1}{8}$ and $x$ .

$\frac{x}{8} {(6 x + 7)}^{\frac{4}{3}} - \frac{1}{48} \cdot \frac{3}{7} u^{\frac{7}{3}} + C$

Step 9.2.2

Combine $\frac{x}{8}$ and ${(6 x + 7)}^{\frac{4}{3}}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{48} \cdot \frac{3}{7} u^{\frac{7}{3}} + C$

Step 9.2.3

Multiply $\frac{3}{7}$ by $\frac{1}{48}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{3}{7 \cdot 48} u^{\frac{7}{3}} + C$

Step 9.2.4

Multiply $7$ by $48$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{3}{336} u^{\frac{7}{3}} + C$

Step 9.2.5

Cancel the common factor of $3$ and $336$ .

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Step 9.2.5.1

Factor $3$ out of $3$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{3 (1)}{336} u^{\frac{7}{3}} + C$

Step 9.2.5.2

Cancel the common factors.

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Step 9.2.5.2.1

Factor $3$ out of $336$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{3 \cdot 1}{3 \cdot 112} u^{\frac{7}{3}} + C$

Step 9.2.5.2.2

Cancel the common factor.

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{3 \cdot 1}{3 \cdot 112} u^{\frac{7}{3}} + C$

Step 9.2.5.2.3

Rewrite the expression.

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{112} u^{\frac{7}{3}} + C$

Step 9.2.6

To write $- \frac{1}{112} u^{\frac{7}{3}}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} - \frac{1}{112} u^{\frac{7}{3}} \cdot \frac{8}{8} + C$

Step 9.2.7

Combine $- \frac{1}{112} u^{\frac{7}{3}}$ and $\frac{8}{8}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}}}{8} + \frac{- \frac{1}{112} u^{\frac{7}{3}} \cdot 8}{8} + C$

Step 9.2.8

Combine the numerators over the common denominator.

$\frac{x {(6 x + 7)}^{\frac{4}{3}} - \frac{1}{112} u^{\frac{7}{3}} \cdot 8}{8} + C$

Step 9.2.9

Combine $u^{\frac{7}{3}}$ and $\frac{1}{112}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}} - \frac{u^{\frac{7}{3}}}{112} \cdot 8}{8} + C$

Step 9.2.10

Multiply $8$ by $- 1$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}} - 8 \frac{u^{\frac{7}{3}}}{112}}{8} + C$

Step 9.2.11

Combine $- 8$ and $\frac{u^{\frac{7}{3}}}{112}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}} + \frac{- 8 u^{\frac{7}{3}}}{112}}{8} + C$

Step 9.2.12

Factor $8$ out of $- 8 u^{\frac{7}{3}}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}} + \frac{8 (- u^{\frac{7}{3}})}{112}}{8} + C$

Step 9.2.13

Cancel the common factors.

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Step 9.2.13.1

Factor $8$ out of $112$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}} + \frac{8 (- u^{\frac{7}{3}})}{8 (14)}}{8} + C$

Step 9.2.13.2

Cancel the common factor.

$\frac{x {(6 x + 7)}^{\frac{4}{3}} + \frac{8 (- u^{\frac{7}{3}})}{8 \cdot 14}}{8} + C$

Step 9.2.13.3

Rewrite the expression.

$\frac{x {(6 x + 7)}^{\frac{4}{3}} + \frac{- u^{\frac{7}{3}}}{14}}{8} + C$

Step 9.2.14

Move the negative in front of the fraction.

$\frac{x {(6 x + 7)}^{\frac{4}{3}} - \frac{u^{\frac{7}{3}}}{14}}{8} + C$

Step 9.2.15

To write $x {(6 x + 7)}^{\frac{4}{3}}$ as a fraction with a common denominator, multiply by $\frac{14}{14}$ .

$\frac{x {(6 x + 7)}^{\frac{4}{3}} \cdot \frac{14}{14} - \frac{u^{\frac{7}{3}}}{14}}{8} + C$

Step 9.2.16

Combine $x {(6 x + 7)}^{\frac{4}{3}}$ and $\frac{14}{14}$ .

$\frac{\frac{x {(6 x + 7)}^{\frac{4}{3}} \cdot 14}{14} - \frac{u^{\frac{7}{3}}}{14}}{8} + C$

Step 9.2.17

Combine the numerators over the common denominator.

$\frac{\frac{x {(6 x + 7)}^{\frac{4}{3}} \cdot 14 - u^{\frac{7}{3}}}{14}}{8} + C$

Step 9.2.18

Move $14$ to the left of $x {(6 x + 7)}^{\frac{4}{3}}$ .

$\frac{\frac{14 \cdot (x {(6 x + 7)}^{\frac{4}{3}}) - u^{\frac{7}{3}}}{14}}{8} + C$

Step 9.2.19

Rewrite $\frac{\frac{14 x {(6 x + 7)}^{\frac{4}{3}} - u^{\frac{7}{3}}}{14}}{8}$ as a product.

$\frac{14 x {(6 x + 7)}^{\frac{4}{3}} - u^{\frac{7}{3}}}{14} \cdot \frac{1}{8} + C$

Step 9.2.20

Multiply $\frac{14 x {(6 x + 7)}^{\frac{4}{3}} - u^{\frac{7}{3}}}{14}$ by $\frac{1}{8}$ .

$\frac{14 x {(6 x + 7)}^{\frac{4}{3}} - u^{\frac{7}{3}}}{14 \cdot 8} + C$

Step 9.2.21

Multiply $14$ by $8$ .

$\frac{14 x {(6 x + 7)}^{\frac{4}{3}} - u^{\frac{7}{3}}}{112} + C$

Step 10

Replace all occurrences of $u$ with $6 x + 7$ .

$\frac{14 x {(6 x + 7)}^{\frac{4}{3}} - {(6 x + 7)}^{\frac{7}{3}}}{112} + C$

Step 11

Reorder terms.

$\frac{1}{112} (14 x {(6 x + 7)}^{\frac{4}{3}} - {(6 x + 7)}^{\frac{7}{3}}) + C$