Calculus Examples

$\int_{0}^{25} \sqrt{x} d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int_{0}^{25} x^{\frac{1}{2}} d x$

Step 2

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

$\frac{2}{3} x^{\frac{3}{2}}]_{0}^{25}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}}$ at $25$ and at $0$ .

$(\frac{2}{3} \cdot 25^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Rewrite $25$ as $5^{2}$ .

$\frac{2}{3} \cdot {(5^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2.2

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

$\frac{2}{3} \cdot 5^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Cancel the common factor.

$\frac{2}{3} \cdot 5^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2.3.2

Rewrite the expression.

$\frac{2}{3} \cdot 5^{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2.4

Raise $5$ to the power of $3$ .

$\frac{2}{3} \cdot 125 - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2.5

Combine $\frac{2}{3}$ and $125$ .

$\frac{2 \cdot 125}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2.6

Multiply $2$ by $125$ .

$\frac{250}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2.7

Rewrite $0$ as $0^{2}$ .

$\frac{250}{3} - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}}$

Step 3.2.8

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

$\frac{250}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}$

Step 3.2.9

Cancel the common factor of $2$ .

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Step 3.2.9.1

Cancel the common factor.

$\frac{250}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}$

Step 3.2.9.2

Rewrite the expression.

$\frac{250}{3} - \frac{2}{3} \cdot 0^{3}$

Step 3.2.10

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

$\frac{250}{3} - \frac{2}{3} \cdot 0$

Step 3.2.11

Multiply $0$ by $- 1$ .

$\frac{250}{3} + 0 (\frac{2}{3})$

Step 3.2.12

Multiply $0$ by $\frac{2}{3}$ .

$\frac{250}{3} + 0$

Step 3.2.13

Add $\frac{250}{3}$ and $0$ .

$\frac{250}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{250}{3}$

Decimal Form:

$83.\overline{3}$

Mixed Number Form:

$83 \frac{1}{3}$

Step 5