Calculus Examples

$\int_{0}^{t^{10}} \sqrt[5]{u} d u$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{u}$ as $u^{\frac{1}{5}}$ .

$\int_{0}^{t^{10}} u^{\frac{1}{5}} d u$

Step 2

By the Power Rule, the integral of $u^{\frac{1}{5}}$ with respect to $u$ is $\frac{5}{6} u^{\frac{6}{5}}$ .

$\frac{5}{6} u^{\frac{6}{5}}]_{0}^{t^{10}}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{5}{6} u^{\frac{6}{5}}$ at $t^{10}$ and at $0$ .

$(\frac{5}{6} {(t^{10})}^{\frac{6}{5}}) - \frac{5}{6} \cdot 0^{\frac{6}{5}}$

Step 3.2

Simplify.

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Step 3.2.1

Multiply the exponents in ${(t^{10})}^{\frac{6}{5}}$ .

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Step 3.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{6} t^{10 (\frac{6}{5})} - \frac{5}{6} \cdot 0^{\frac{6}{5}}$

Step 3.2.1.2

Cancel the common factor of $5$ .

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Step 3.2.1.2.1

Factor $5$ out of $10$ .

$\frac{5}{6} t^{5 (2) \frac{6}{5}} - \frac{5}{6} \cdot 0^{\frac{6}{5}}$

Step 3.2.1.2.2

Cancel the common factor.

$\frac{5}{6} t^{5 \cdot 2 \frac{6}{5}} - \frac{5}{6} \cdot 0^{\frac{6}{5}}$

Step 3.2.1.2.3

Rewrite the expression.

$\frac{5}{6} t^{2 \cdot 6} - \frac{5}{6} \cdot 0^{\frac{6}{5}}$

Step 3.2.1.3

Multiply $2$ by $6$ .

$\frac{5}{6} t^{12} - \frac{5}{6} \cdot 0^{\frac{6}{5}}$

Step 3.2.2

Rewrite $0$ as $0^{5}$ .

$\frac{5}{6} t^{12} - \frac{5}{6} \cdot {(0^{5})}^{\frac{6}{5}}$

Step 3.2.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{6} t^{12} - \frac{5}{6} \cdot 0^{5 (\frac{6}{5})}$

Step 3.2.4

Cancel the common factor of $5$ .

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Step 3.2.4.1

Cancel the common factor.

$\frac{5}{6} t^{12} - \frac{5}{6} \cdot 0^{5 (\frac{6}{5})}$

Step 3.2.4.2

Rewrite the expression.

$\frac{5}{6} t^{12} - \frac{5}{6} \cdot 0^{6}$

Step 3.2.5

Raising $0$ to any positive power yields $0$ .

$\frac{5}{6} t^{12} - \frac{5}{6} \cdot 0$

Step 3.2.6

Multiply $0$ by $- 1$ .

$\frac{5}{6} t^{12} + 0 (\frac{5}{6})$

Step 3.2.7

Multiply $0$ by $\frac{5}{6}$ .

$\frac{5}{6} t^{12} + 0$

Step 3.2.8

Add $\frac{5}{6} t^{12}$ and $0$ .

$\frac{5}{6} t^{12}$

Step 4

Combine $\frac{5}{6}$ and $t^{12}$ .

$\frac{5 t^{12}}{6}$