Calculus Examples

$\int x \sqrt[3]{3 - 2 x} d x$

Step 1

Let $u = 3 - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $3 - 2 x$ .

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

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

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

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

Step 1.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2 x]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$0 - 2 \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - 2 \cdot 1$

Step 1.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 1.1.4

Subtract $2$ from $0$ .

$- 2$

Step 1.2

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

$\int (- \frac{u}{2} + \frac{3}{2}) \sqrt[3]{u} \frac{1}{- 2} d u$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Move the negative in front of the fraction.

$\int (- \frac{u}{2} + \frac{3}{2}) \sqrt[3]{u} (- \frac{1}{2}) d u$

Step 2.2

Combine $\sqrt[3]{u}$ and $\frac{1}{2}$ .

$\int (- \frac{u}{2} + \frac{3}{2}) (- \frac{\sqrt[3]{u}}{2}) d u$

Step 3

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

$- \int (- \frac{u}{2} + \frac{3}{2}) (\frac{\sqrt[3]{u}}{2}) d u$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{u}$ as $u^{\frac{1}{3}}$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Apply the distributive property.

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

Step 5.2

Combine $- \frac{u}{2}$ and $\frac{u^{\frac{1}{3}}}{2}$ .

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Write $1$ as a fraction with a common denominator.

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

Step 5.6

Combine the numerators over the common denominator.

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

Step 5.7

Add $3$ and $1$ .

$- \int - \frac{u^{\frac{4}{3}}}{2 \cdot 2} + \frac{3}{2} \cdot \frac{u^{\frac{1}{3}}}{2} d u$

Step 5.8

Multiply $2$ by $2$ .

$- \int - \frac{u^{\frac{4}{3}}}{4} + \frac{3}{2} \cdot \frac{u^{\frac{1}{3}}}{2} d u$

Step 5.9

Multiply $\frac{3}{2}$ by $\frac{u^{\frac{1}{3}}}{2}$ .

$- \int - \frac{u^{\frac{4}{3}}}{4} + \frac{3 u^{\frac{1}{3}}}{2 \cdot 2} d u$

Step 5.10

Multiply $2$ by $2$ .

$- \int - \frac{u^{\frac{4}{3}}}{4} + \frac{3 u^{\frac{1}{3}}}{4} d u$

Step 5.11

Reorder $- \frac{u^{\frac{4}{3}}}{4}$ and $\frac{3 u^{\frac{1}{3}}}{4}$ .

$- \int \frac{3 u^{\frac{1}{3}}}{4} - \frac{u^{\frac{4}{3}}}{4} d u$

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

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

Step 12

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

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

Step 13

Simplify.

Tap for more steps...

Step 13.1

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

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

Step 13.2

Simplify.

$- (\frac{9 u^{\frac{4}{3}}}{16} - \frac{3 u^{\frac{7}{3}}}{28}) + C$

Step 14

Reorder terms.

$- (\frac{9}{16} u^{\frac{4}{3}} - \frac{3}{28} u^{\frac{7}{3}}) + C$

Step 15

Replace all occurrences of $u$ with $3 - 2 x$ .

$- (\frac{9}{16} {(3 - 2 x)}^{\frac{4}{3}} - \frac{3}{28} {(3 - 2 x)}^{\frac{7}{3}}) + C$

Step 16

Simplify.

Tap for more steps...

Step 16.1

Simplify each term.

Tap for more steps...

Step 16.1.1

Combine $\frac{9}{16}$ and ${(3 - 2 x)}^{\frac{4}{3}}$ .

$- (\frac{9 {(3 - 2 x)}^{\frac{4}{3}}}{16} - \frac{3}{28} {(3 - 2 x)}^{\frac{7}{3}}) + C$

Step 16.1.2

Combine ${(3 - 2 x)}^{\frac{7}{3}}$ and $\frac{3}{28}$ .

$- (\frac{9 {(3 - 2 x)}^{\frac{4}{3}}}{16} - \frac{{(3 - 2 x)}^{\frac{7}{3}} \cdot 3}{28}) + C$

Step 16.1.3

Move $3$ to the left of ${(3 - 2 x)}^{\frac{7}{3}}$ .

$- (\frac{9 {(3 - 2 x)}^{\frac{4}{3}}}{16} - \frac{3 {(3 - 2 x)}^{\frac{7}{3}}}{28}) + C$

Step 16.2

To write $\frac{9 {(3 - 2 x)}^{\frac{4}{3}}}{16}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$- (\frac{9 {(3 - 2 x)}^{\frac{4}{3}}}{16} \cdot \frac{7}{7} - \frac{3 {(3 - 2 x)}^{\frac{7}{3}}}{28}) + C$

Step 16.3

To write $- \frac{3 {(3 - 2 x)}^{\frac{7}{3}}}{28}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- (\frac{9 {(3 - 2 x)}^{\frac{4}{3}}}{16} \cdot \frac{7}{7} - \frac{3 {(3 - 2 x)}^{\frac{7}{3}}}{28} \cdot \frac{4}{4}) + C$

Step 16.4

Write each expression with a common denominator of $112$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 16.4.1

Multiply $\frac{9 {(3 - 2 x)}^{\frac{4}{3}}}{16}$ by $\frac{7}{7}$ .

$- (\frac{9 {(3 - 2 x)}^{\frac{4}{3}} \cdot 7}{16 \cdot 7} - \frac{3 {(3 - 2 x)}^{\frac{7}{3}}}{28} \cdot \frac{4}{4}) + C$

Step 16.4.2

Multiply $16$ by $7$ .

$- (\frac{9 {(3 - 2 x)}^{\frac{4}{3}} \cdot 7}{112} - \frac{3 {(3 - 2 x)}^{\frac{7}{3}}}{28} \cdot \frac{4}{4}) + C$

Step 16.4.3

Multiply $\frac{3 {(3 - 2 x)}^{\frac{7}{3}}}{28}$ by $\frac{4}{4}$ .

$- (\frac{9 {(3 - 2 x)}^{\frac{4}{3}} \cdot 7}{112} - \frac{3 {(3 - 2 x)}^{\frac{7}{3}} \cdot 4}{28 \cdot 4}) + C$

Step 16.4.4

Multiply $28$ by $4$ .

$- (\frac{9 {(3 - 2 x)}^{\frac{4}{3}} \cdot 7}{112} - \frac{3 {(3 - 2 x)}^{\frac{7}{3}} \cdot 4}{112}) + C$

Step 16.5

Combine the numerators over the common denominator.

$- \frac{9 {(3 - 2 x)}^{\frac{4}{3}} \cdot 7 - 3 {(3 - 2 x)}^{\frac{7}{3}} \cdot 4}{112} + C$

Step 16.6

Simplify the numerator.

Tap for more steps...

Step 16.6.1

Multiply $7$ by $9$ .

$- \frac{63 {(3 - 2 x)}^{\frac{4}{3}} - 3 {(3 - 2 x)}^{\frac{7}{3}} \cdot 4}{112} + C$

Step 16.6.2

Multiply $4$ by $- 3$ .

$- \frac{63 {(3 - 2 x)}^{\frac{4}{3}} - 12 {(3 - 2 x)}^{\frac{7}{3}}}{112} + C$

Step 17

Reorder terms.

$- \frac{1}{112} (63 {(3 - 2 x)}^{\frac{4}{3}} - 12 {(3 - 2 x)}^{\frac{7}{3}}) + C$