Calculus Examples

$\int_{0}^{1} \sqrt{x} d x$

Step 1

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

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

Step 2

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

$\frac{2}{3} x^{\frac{3}{2}}]_{0}^{1}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}}$ at $1$ and at $0$ .

$(\frac{2}{3} \cdot 1^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

$\frac{2}{3} \cdot 1 - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 3.2.2

Multiply $\frac{2}{3}$ by $1$ .

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Cancel the common factor of $2$ .

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Step 3.2.5.1

Cancel the common factor.

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

Step 3.2.5.2

Rewrite the expression.

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

Step 3.2.6

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

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

Step 3.2.7

Multiply $0$ by $- 1$ .

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

Step 3.2.8

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

$\frac{2}{3} + 0$

Step 3.2.9

Add $\frac{2}{3}$ and $0$ .

$\frac{2}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$

Step 5