Calculus Examples

$\int_{0}^{1} x^{2} + \sqrt{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} x^{2} d x + \int_{0}^{1} \sqrt{x} d x$

Step 2

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

$\frac{1}{3} x^{3}]_{0}^{1} + \int_{0}^{1} \sqrt{x} d x$

Step 3

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

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

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

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

Step 5.2.2

Simplify.

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Step 5.2.2.1

One to any power is one.

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

Step 5.2.2.2

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

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

Step 5.2.2.3

One to any power is one.

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

Step 5.2.2.4

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

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

Step 5.2.2.5

Combine the numerators over the common denominator.

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

Step 5.2.2.6

Add $1$ and $2$ .

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

Step 5.2.2.7

Cancel the common factor of $3$ .

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Step 5.2.2.7.1

Cancel the common factor.

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

Step 5.2.2.7.2

Rewrite the expression.

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

Step 5.2.2.8

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

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

Step 5.2.2.9

Multiply $\frac{1}{3}$ by $0$ .

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

Step 5.2.2.10

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

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

Step 5.2.2.11

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

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

Step 5.2.2.12

Cancel the common factor of $2$ .

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Step 5.2.2.12.1

Cancel the common factor.

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

Step 5.2.2.12.2

Rewrite the expression.

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

Step 5.2.2.13

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

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

Step 5.2.2.14

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

$1 - (0 + 0)$

Step 5.2.2.15

Add $0$ and $0$ .

$1 - 0$

Step 5.2.2.16

Multiply $- 1$ by $0$ .

$1 + 0$

Step 5.2.2.17

Add $1$ and $0$ .

$1$

Step 6