Calculus Examples

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

Step 1

Since $5$ is constant with respect to $u$ , move $5$ out of the integral.

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

Step 2

Let $u_{1} = 3 + 2 u^{3}$ . Then $d u_{1} = 6 u^{2} d u$ , so $\frac{1}{6} d u_{1} = u^{2} d u$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.1

Let $u_{1} = 3 + 2 u^{3}$ . Find $\frac{d u_{1}}{d u}$ .

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Step 2.1.1

Differentiate $3 + 2 u^{3}$ .

$\frac{d}{d u} [3 + 2 u^{3}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

$\frac{d}{d u} [3] + \frac{d}{d u} [2 u^{3}]$

Step 2.1.2.2

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

$0 + \frac{d}{d u} [2 u^{3}]$

Step 2.1.3

Evaluate $\frac{d}{d u} [2 u^{3}]$ .

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Step 2.1.3.1

Since $2$ is constant with respect to $u$ , the derivative of $2 u^{3}$ with respect to $u$ is $2 \frac{d}{d u} [u^{3}]$ .

$0 + 2 \frac{d}{d u} [u^{3}]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$0 + 2 (3 u^{2})$

Step 2.1.3.3

Multiply $3$ by $2$ .

$0 + 6 u^{2}$

Step 2.1.4

Add $0$ and $6 u^{2}$ .

$6 u^{2}$

Step 2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$5 \int {u_{1}}^{\frac{2}{3}} \frac{1}{6} d u_{1}$

Step 3

Combine ${u_{1}}^{\frac{2}{3}}$ and $\frac{1}{6}$ .

$5 \int \frac{{u_{1}}^{\frac{2}{3}}}{6} d u_{1}$

Step 4

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

$5 (\frac{1}{6} \int {u_{1}}^{\frac{2}{3}} d u_{1})$

Step 5

Combine $\frac{1}{6}$ and $5$ .

$\frac{5}{6} \int {u_{1}}^{\frac{2}{3}} d u_{1}$

Step 6

By the Power Rule, the integral of ${u_{1}}^{\frac{2}{3}}$ with respect to $u_{1}$ is $\frac{3}{5} {u_{1}}^{\frac{5}{3}}$ .

$\frac{5}{6} (\frac{3}{5} {u_{1}}^{\frac{5}{3}} + C)$

Step 7

Simplify.

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Step 7.1

Rewrite $\frac{5}{6} (\frac{3}{5} {u_{1}}^{\frac{5}{3}} + C)$ as $\frac{5}{6} \cdot \frac{3}{5} {u_{1}}^{\frac{5}{3}} + C$ .

$\frac{5}{6} \cdot \frac{3}{5} {u_{1}}^{\frac{5}{3}} + C$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $\frac{5}{6}$ by $\frac{3}{5}$ .

$\frac{5 \cdot 3}{6 \cdot 5} {u_{1}}^{\frac{5}{3}} + C$

Step 7.2.2

Multiply $5$ by $3$ .

$\frac{15}{6 \cdot 5} {u_{1}}^{\frac{5}{3}} + C$

Step 7.2.3

Multiply $6$ by $5$ .

$\frac{15}{30} {u_{1}}^{\frac{5}{3}} + C$

Step 7.2.4

Cancel the common factor of $15$ and $30$ .

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Step 7.2.4.1

Factor $15$ out of $15$ .

$\frac{15 (1)}{30} {u_{1}}^{\frac{5}{3}} + C$

Step 7.2.4.2

Cancel the common factors.

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Step 7.2.4.2.1

Factor $15$ out of $30$ .

$\frac{15 \cdot 1}{15 \cdot 2} {u_{1}}^{\frac{5}{3}} + C$

Step 7.2.4.2.2

Cancel the common factor.

$\frac{15 \cdot 1}{15 \cdot 2} {u_{1}}^{\frac{5}{3}} + C$

Step 7.2.4.2.3

Rewrite the expression.

$\frac{1}{2} {u_{1}}^{\frac{5}{3}} + C$

Step 8

Replace all occurrences of $u_{1}$ with $3 + 2 u^{3}$ .

$\frac{1}{2} {(3 + 2 u^{3})}^{\frac{5}{3}} + C$