Calculus Examples

$\int_{0}^{1} 13 \sqrt[13]{x} d x$

Step 1

Since $13$ is constant with respect to $x$ , move $13$ out of the integral.

$13 \int_{0}^{1} \sqrt[13]{x} d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[13]{x}$ as $x^{\frac{1}{13}}$ .

$13 \int_{0}^{1} x^{\frac{1}{13}} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{1}{13}}$ with respect to $x$ is $\frac{13}{14} x^{\frac{14}{13}}$ .

$13 (\frac{13}{14} x^{\frac{14}{13}}]_{0}^{1})$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Combine $\frac{13}{14}$ and $x^{\frac{14}{13}}$ .

$13 (\frac{13 x^{\frac{14}{13}}}{14}]_{0}^{1})$

Step 4.2

Substitute and simplify.

Tap for more steps...

Step 4.2.1

Evaluate $\frac{13 x^{\frac{14}{13}}}{14}$ at $1$ and at $0$ .

$13 ((\frac{13 \cdot 1^{\frac{14}{13}}}{14}) - \frac{13 \cdot 0^{\frac{14}{13}}}{14})$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

One to any power is one.

$13 (\frac{13 \cdot 1}{14} - \frac{13 \cdot 0^{\frac{14}{13}}}{14})$

Step 4.2.2.2

Multiply $13$ by $1$ .

$13 (\frac{13}{14} - \frac{13 \cdot 0^{\frac{14}{13}}}{14})$

Step 4.2.2.3

Rewrite $0$ as $0^{13}$ .

$13 (\frac{13}{14} - \frac{13 \cdot {(0^{13})}^{\frac{14}{13}}}{14})$

Step 4.2.2.4

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

$13 (\frac{13}{14} - \frac{13 \cdot 0^{13 (\frac{14}{13})}}{14})$

Step 4.2.2.5

Cancel the common factor of $13$ .

Tap for more steps...

Step 4.2.2.5.1

Cancel the common factor.

$13 (\frac{13}{14} - \frac{13 \cdot 0^{13 (\frac{14}{13})}}{14})$

Step 4.2.2.5.2

Rewrite the expression.

$13 (\frac{13}{14} - \frac{13 \cdot 0^{14}}{14})$

Step 4.2.2.6

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

$13 (\frac{13}{14} - \frac{13 \cdot 0}{14})$

Step 4.2.2.7

Multiply $13$ by $0$ .

$13 (\frac{13}{14} - \frac{0}{14})$

Step 4.2.2.8

Cancel the common factor of $0$ and $14$ .

Tap for more steps...

Step 4.2.2.8.1

Factor $14$ out of $0$ .

$13 (\frac{13}{14} - \frac{14 (0)}{14})$

Step 4.2.2.8.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.8.2.1

Factor $14$ out of $14$ .

$13 (\frac{13}{14} - \frac{14 \cdot 0}{14 \cdot 1})$

Step 4.2.2.8.2.2

Cancel the common factor.

$13 (\frac{13}{14} - \frac{14 \cdot 0}{14 \cdot 1})$

Step 4.2.2.8.2.3

Rewrite the expression.

$13 (\frac{13}{14} - \frac{0}{1})$

Step 4.2.2.8.2.4

Divide $0$ by $1$ .

$13 (\frac{13}{14} - 0)$

Step 4.2.2.9

Multiply $- 1$ by $0$ .

$13 (\frac{13}{14} + 0)$

Step 4.2.2.10

Add $\frac{13}{14}$ and $0$ .

$13 (\frac{13}{14})$

Step 4.2.2.11

Combine $13$ and $\frac{13}{14}$ .

$\frac{13 \cdot 13}{14}$

Step 4.2.2.12

Multiply $13$ by $13$ .

$\frac{169}{14}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{169}{14}$

Decimal Form:

$12.07142857 \dots$

Mixed Number Form:

$12 \frac{1}{14}$

Step 6