Calculus Examples

$\frac{10}{3 \int {(1 + 0.2 t)}^{\frac{3}{2}} d t}$

Step 1

Let $u = 1 + 0.2 t$ . Then $d u = 0.2 d t$ , so $\frac{1}{0.2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 + 0.2 t$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $1 + 0.2 t$ .

$\frac{d}{d t} [1 + 0.2 t]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 + 0.2 t$ with respect to $t$ is $\frac{d}{d t} [1] + \frac{d}{d t} [0.2 t]$ .

$\frac{d}{d t} [1] + \frac{d}{d t} [0.2 t]$

Step 1.1.2.2

Since $1$ is constant with respect to $t$ , the derivative of $1$ with respect to $t$ is $0$ .

$0 + \frac{d}{d t} [0.2 t]$

Step 1.1.3

Evaluate $\frac{d}{d t} [0.2 t]$ .

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Step 1.1.3.1

Since $0.2$ is constant with respect to $t$ , the derivative of $0.2 t$ with respect to $t$ is $0.2 \frac{d}{d t} [t]$ .

$0 + 0.2 \frac{d}{d t} [t]$

Step 1.1.3.2

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

$0 + 0.2 \cdot 1$

Step 1.1.3.3

Multiply $0.2$ by $1$ .

$0 + 0.2$

Step 1.1.4

Add $0$ and $0.2$ .

$0.2$

Step 1.2

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

$\frac{10}{3 \int u^{\frac{3}{2}} \frac{1}{0.2} d u}$

Step 2

Combine $u^{\frac{3}{2}}$ and $\frac{1}{0.2}$ .

$\frac{10}{3 \int \frac{u^{\frac{3}{2}}}{0.2} d u}$

Step 3

Since $\frac{1}{0.2}$ is constant with respect to $u$ , move $\frac{1}{0.2}$ out of the integral.

$\frac{10}{3 (\frac{1}{0.2} \int u^{\frac{3}{2}} d u)}$

Step 4

Combine $\frac{1}{0.2}$ and $3$ .

$\frac{10}{\frac{3}{0.2} \int u^{\frac{3}{2}} d u}$

Step 5

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

$\frac{10}{\frac{3}{0.2} (\frac{2}{5} u^{\frac{5}{2}} + C)}$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{10}{\frac{3}{0.2} (\frac{2}{5} u^{\frac{5}{2}} + C)}$ as $\frac{10}{15 (\frac{2}{5} u^{\frac{5}{2}})} + C$ .

$\frac{10}{15 (\frac{2}{5} u^{\frac{5}{2}})} + C$

Step 6.2

Simplify.

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Step 6.2.1

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

$\frac{10}{15 \frac{2 u^{\frac{5}{2}}}{5}} + C$

Step 6.2.2

Combine $15$ and $\frac{2 u^{\frac{5}{2}}}{5}$ .

$\frac{10}{\frac{15 (2 u^{\frac{5}{2}})}{5}} + C$

Step 6.2.3

Multiply $2$ by $15$ .

$\frac{10}{\frac{30 u^{\frac{5}{2}}}{5}} + C$

Step 6.2.4

Factor $5$ out of $30 u^{\frac{5}{2}}$ .

$\frac{10}{\frac{5 (6 u^{\frac{5}{2}})}{5}} + C$

Step 6.2.5

Cancel the common factors.

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Step 6.2.5.1

Factor $5$ out of $5$ .

$\frac{10}{\frac{5 (6 u^{\frac{5}{2}})}{5 (1)}} + C$

Step 6.2.5.2

Cancel the common factor.

$\frac{10}{\frac{5 (6 u^{\frac{5}{2}})}{5 \cdot 1}} + C$

Step 6.2.5.3

Rewrite the expression.

$\frac{10}{\frac{6 u^{\frac{5}{2}}}{1}} + C$

Step 6.2.5.4

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

$\frac{10}{6 u^{\frac{5}{2}}} + C$

Step 6.2.6

Factor $2$ out of $10$ .

$\frac{2 (5)}{6 u^{\frac{5}{2}}} + C$

Step 6.2.7

Cancel the common factors.

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Step 6.2.7.1

Factor $2$ out of $6 u^{\frac{5}{2}}$ .

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

Step 6.2.7.2

Cancel the common factor.

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

Step 6.2.7.3

Rewrite the expression.

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

Step 7

Replace all occurrences of $u$ with $1 + 0.2 t$ .

$\frac{5}{3 {(1 + 0.2 t)}^{\frac{5}{2}}} + C$