Calculus Examples

$\int x \sqrt[3]{x + 5} d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = \sqrt[3]{x + 5}$ .

$x (\frac{3}{4} {(x + 5)}^{\frac{4}{3}}) - \int \frac{3}{4} {(x + 5)}^{\frac{4}{3}} d x$

Step 2

Simplify.

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Step 2.1

Combine $\frac{3}{4}$ and ${(x + 5)}^{\frac{4}{3}}$ .

$x \frac{3 {(x + 5)}^{\frac{4}{3}}}{4} - \int \frac{3}{4} {(x + 5)}^{\frac{4}{3}} d x$

Step 2.2

Combine $x$ and $\frac{3 {(x + 5)}^{\frac{4}{3}}}{4}$ .

$\frac{x (3 {(x + 5)}^{\frac{4}{3}})}{4} - \int \frac{3}{4} {(x + 5)}^{\frac{4}{3}} d x$

Step 3

Since $\frac{3}{4}$ is constant with respect to $x$ , move $\frac{3}{4}$ out of the integral.

$\frac{x (3 {(x + 5)}^{\frac{4}{3}})}{4} - (\frac{3}{4} \int {(x + 5)}^{\frac{4}{3}} d x)$

Step 4

Let $u = x + 5$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

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

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Step 4.1.1

Differentiate $x + 5$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

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

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

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

$\frac{x (3 {(x + 5)}^{\frac{4}{3}})}{4} - \frac{3}{4} \int u^{\frac{4}{3}} d u$

Step 5

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

$\frac{x (3 {(x + 5)}^{\frac{4}{3}})}{4} - \frac{3}{4} (\frac{3}{7} u^{\frac{7}{3}} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{x (3 {(x + 5)}^{\frac{4}{3}})}{4} - \frac{3}{4} (\frac{3}{7} u^{\frac{7}{3}} + C)$ as $\frac{3}{4} x {(x + 5)}^{\frac{4}{3}} - \frac{3}{4} \cdot \frac{3}{7} u^{\frac{7}{3}} + C$ .

$\frac{3}{4} x {(x + 5)}^{\frac{4}{3}} - \frac{3}{4} \cdot \frac{3}{7} u^{\frac{7}{3}} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $\frac{3}{4}$ and $x$ .

$\frac{3 x}{4} {(x + 5)}^{\frac{4}{3}} - \frac{3}{4} \cdot \frac{3}{7} u^{\frac{7}{3}} + C$

Step 6.2.2

Combine $\frac{3 x}{4}$ and ${(x + 5)}^{\frac{4}{3}}$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}}}{4} - \frac{3}{4} \cdot \frac{3}{7} u^{\frac{7}{3}} + C$

Step 6.2.3

Multiply $\frac{3}{7}$ by $\frac{3}{4}$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}}}{4} - \frac{3 \cdot 3}{7 \cdot 4} u^{\frac{7}{3}} + C$

Step 6.2.4

Multiply $3$ by $3$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}}}{4} - \frac{9}{7 \cdot 4} u^{\frac{7}{3}} + C$

Step 6.2.5

Multiply $7$ by $4$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}}}{4} - \frac{9}{28} u^{\frac{7}{3}} + C$

Step 6.2.6

To write $- \frac{9}{28} u^{\frac{7}{3}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}}}{4} - \frac{9}{28} u^{\frac{7}{3}} \cdot \frac{4}{4} + C$

Step 6.2.7

Combine $- \frac{9}{28} u^{\frac{7}{3}}$ and $\frac{4}{4}$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}}}{4} + \frac{- \frac{9}{28} u^{\frac{7}{3}} \cdot 4}{4} + C$

Step 6.2.8

Combine the numerators over the common denominator.

$\frac{3 x {(x + 5)}^{\frac{4}{3}} - \frac{9}{28} u^{\frac{7}{3}} \cdot 4}{4} + C$

Step 6.2.9

Combine $u^{\frac{7}{3}}$ and $\frac{9}{28}$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}} - \frac{u^{\frac{7}{3}} \cdot 9}{28} \cdot 4}{4} + C$

Step 6.2.10

Multiply $4$ by $- 1$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}} - 4 \frac{u^{\frac{7}{3}} \cdot 9}{28}}{4} + C$

Step 6.2.11

Combine $- 4$ and $\frac{u^{\frac{7}{3}} \cdot 9}{28}$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}} + \frac{- 4 (u^{\frac{7}{3}} \cdot 9)}{28}}{4} + C$

Step 6.2.12

Multiply $9$ by $- 4$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}} + \frac{- 36 u^{\frac{7}{3}}}{28}}{4} + C$

Step 6.2.13

Factor $4$ out of $- 36 u^{\frac{7}{3}}$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}} + \frac{4 (- 9 u^{\frac{7}{3}})}{28}}{4} + C$

Step 6.2.14

Cancel the common factors.

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Step 6.2.14.1

Factor $4$ out of $28$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}} + \frac{4 (- 9 u^{\frac{7}{3}})}{4 (7)}}{4} + C$

Step 6.2.14.2

Cancel the common factor.

$\frac{3 x {(x + 5)}^{\frac{4}{3}} + \frac{4 (- 9 u^{\frac{7}{3}})}{4 \cdot 7}}{4} + C$

Step 6.2.14.3

Rewrite the expression.

$\frac{3 x {(x + 5)}^{\frac{4}{3}} + \frac{- 9 u^{\frac{7}{3}}}{7}}{4} + C$

Step 6.2.15

Move the negative in front of the fraction.

$\frac{3 x {(x + 5)}^{\frac{4}{3}} - \frac{9 u^{\frac{7}{3}}}{7}}{4} + C$

Step 6.2.16

To write $3 x {(x + 5)}^{\frac{4}{3}}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{3 x {(x + 5)}^{\frac{4}{3}} \cdot \frac{7}{7} - \frac{9 u^{\frac{7}{3}}}{7}}{4} + C$

Step 6.2.17

Combine $3 x {(x + 5)}^{\frac{4}{3}}$ and $\frac{7}{7}$ .

$\frac{\frac{3 x {(x + 5)}^{\frac{4}{3}} \cdot 7}{7} - \frac{9 u^{\frac{7}{3}}}{7}}{4} + C$

Step 6.2.18

Combine the numerators over the common denominator.

$\frac{\frac{3 x {(x + 5)}^{\frac{4}{3}} \cdot 7 - 9 u^{\frac{7}{3}}}{7}}{4} + C$

Step 6.2.19

Multiply $7$ by $3$ .

$\frac{\frac{21 x {(x + 5)}^{\frac{4}{3}} - 9 u^{\frac{7}{3}}}{7}}{4} + C$

Step 6.2.20

Rewrite $\frac{\frac{21 x {(x + 5)}^{\frac{4}{3}} - 9 u^{\frac{7}{3}}}{7}}{4}$ as a product.

$\frac{21 x {(x + 5)}^{\frac{4}{3}} - 9 u^{\frac{7}{3}}}{7} \cdot \frac{1}{4} + C$

Step 6.2.21

Multiply $\frac{21 x {(x + 5)}^{\frac{4}{3}} - 9 u^{\frac{7}{3}}}{7}$ by $\frac{1}{4}$ .

$\frac{21 x {(x + 5)}^{\frac{4}{3}} - 9 u^{\frac{7}{3}}}{7 \cdot 4} + C$

Step 6.2.22

Multiply $7$ by $4$ .

$\frac{21 x {(x + 5)}^{\frac{4}{3}} - 9 u^{\frac{7}{3}}}{28} + C$

Step 7

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

$\frac{21 x {(x + 5)}^{\frac{4}{3}} - 9 {(x + 5)}^{\frac{7}{3}}}{28} + C$

Step 8

Reorder terms.

$\frac{1}{28} (21 x {(x + 5)}^{\frac{4}{3}} - 9 {(x + 5)}^{\frac{7}{3}}) + C$