Calculus Examples

$\int_{0}^{1} 11 \sqrt[11]{x} d x$

Step 1

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$11 \int_{0}^{1} \sqrt[11]{x} d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[11]{x}$ as $x^{\frac{1}{11}}$ .

$11 \int_{0}^{1} x^{\frac{1}{11}} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{1}{11}}$ with respect to $x$ is $\frac{11}{12} x^{\frac{12}{11}}$ .

$11 (\frac{11}{12} x^{\frac{12}{11}}]_{0}^{1})$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{11}{12}$ and $x^{\frac{12}{11}}$ .

$11 (\frac{11 x^{\frac{12}{11}}}{12}]_{0}^{1})$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{11 x^{\frac{12}{11}}}{12}$ at $1$ and at $0$ .

$11 ((\frac{11 \cdot 1^{\frac{12}{11}}}{12}) - \frac{11 \cdot 0^{\frac{12}{11}}}{12})$

Step 4.2.2

Simplify.

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Step 4.2.2.1

One to any power is one.

$11 (\frac{11 \cdot 1}{12} - \frac{11 \cdot 0^{\frac{12}{11}}}{12})$

Step 4.2.2.2

Multiply $11$ by $1$ .

$11 (\frac{11}{12} - \frac{11 \cdot 0^{\frac{12}{11}}}{12})$

Step 4.2.2.3

Rewrite $0$ as $0^{11}$ .

$11 (\frac{11}{12} - \frac{11 \cdot {(0^{11})}^{\frac{12}{11}}}{12})$

Step 4.2.2.4

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

$11 (\frac{11}{12} - \frac{11 \cdot 0^{11 (\frac{12}{11})}}{12})$

Step 4.2.2.5

Cancel the common factor of $11$ .

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Step 4.2.2.5.1

Cancel the common factor.

$11 (\frac{11}{12} - \frac{11 \cdot 0^{11 (\frac{12}{11})}}{12})$

Step 4.2.2.5.2

Rewrite the expression.

$11 (\frac{11}{12} - \frac{11 \cdot 0^{12}}{12})$

Step 4.2.2.6

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

$11 (\frac{11}{12} - \frac{11 \cdot 0}{12})$

Step 4.2.2.7

Multiply $11$ by $0$ .

$11 (\frac{11}{12} - \frac{0}{12})$

Step 4.2.2.8

Cancel the common factor of $0$ and $12$ .

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Step 4.2.2.8.1

Factor $12$ out of $0$ .

$11 (\frac{11}{12} - \frac{12 (0)}{12})$

Step 4.2.2.8.2

Cancel the common factors.

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Step 4.2.2.8.2.1

Factor $12$ out of $12$ .

$11 (\frac{11}{12} - \frac{12 \cdot 0}{12 \cdot 1})$

Step 4.2.2.8.2.2

Cancel the common factor.

$11 (\frac{11}{12} - \frac{12 \cdot 0}{12 \cdot 1})$

Step 4.2.2.8.2.3

Rewrite the expression.

$11 (\frac{11}{12} - \frac{0}{1})$

Step 4.2.2.8.2.4

Divide $0$ by $1$ .

$11 (\frac{11}{12} - 0)$

Step 4.2.2.9

Multiply $- 1$ by $0$ .

$11 (\frac{11}{12} + 0)$

Step 4.2.2.10

Add $\frac{11}{12}$ and $0$ .

$11 (\frac{11}{12})$

Step 4.2.2.11

Combine $11$ and $\frac{11}{12}$ .

$\frac{11 \cdot 11}{12}$

Step 4.2.2.12

Multiply $11$ by $11$ .

$\frac{121}{12}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{121}{12}$

Decimal Form:

$10.08\overline{3}$

Mixed Number Form:

$10 \frac{1}{12}$

Step 6