Calculus Examples

$\int_{0}^{1} 5 x^{2} + \sqrt{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} 5 x^{2} d x + \int_{0}^{1} \sqrt{x} d x$

Step 2

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{0}^{1} x^{2} d x + \int_{0}^{1} \sqrt{x} d x$

Step 3

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

$5 (\frac{1}{3} x^{3}]_{0}^{1}) + \int_{0}^{1} \sqrt{x} d x$

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$5 (\frac{x^{3}}{3}]_{0}^{1}) + \int_{0}^{1} \sqrt{x} d x$

Step 4.2

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

$5 (\frac{x^{3}}{3}]_{0}^{1}) + \int_{0}^{1} x^{\frac{1}{2}} d x$

Step 5

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

$5 (\frac{x^{3}}{3}]_{0}^{1}) + \frac{2}{3} x^{\frac{3}{2}}]_{0}^{1}$

Step 6

Substitute and simplify.

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Step 6.1

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

$5 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}) + \frac{2}{3} x^{\frac{3}{2}}]_{0}^{1}$

Step 6.2

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

$5 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3

Simplify.

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Step 6.3.1

One to any power is one.

$5 (\frac{1}{3} - \frac{0^{3}}{3}) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.2

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

$5 (\frac{1}{3} - \frac{0}{3}) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.3

Cancel the common factor of $0$ and $3$ .

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Step 6.3.3.1

Factor $3$ out of $0$ .

$5 (\frac{1}{3} - \frac{3 (0)}{3}) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $3$ out of $3$ .

$5 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.3.2.2

Cancel the common factor.

$5 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.3.2.3

Rewrite the expression.

$5 (\frac{1}{3} - \frac{0}{1}) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.3.2.4

Divide $0$ by $1$ .

$5 (\frac{1}{3} - 0) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.4

Multiply $- 1$ by $0$ .

$5 (\frac{1}{3} + 0) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.5

Add $\frac{1}{3}$ and $0$ .

$5 (\frac{1}{3}) + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.6

Combine $5$ and $\frac{1}{3}$ .

$\frac{5}{3} + \frac{2}{3} \cdot 1^{\frac{3}{2}} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.7

One to any power is one.

$\frac{5}{3} + \frac{2}{3} \cdot 1 - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 6.3.8

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

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

Step 6.3.9

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

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

Step 6.3.10

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

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

Step 6.3.11

Cancel the common factor of $2$ .

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Step 6.3.11.1

Cancel the common factor.

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

Step 6.3.11.2

Rewrite the expression.

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

Step 6.3.12

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

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

Step 6.3.13

Multiply $0$ by $- 1$ .

$\frac{5}{3} + \frac{2}{3} + 0 (\frac{2}{3})$

Step 6.3.14

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

$\frac{5}{3} + \frac{2}{3} + 0$

Step 6.3.15

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

$\frac{5}{3} + \frac{2}{3}$

Step 6.3.16

Combine the numerators over the common denominator.

$\frac{5 + 2}{3}$

Step 6.3.17

Add $5$ and $2$ .

$\frac{7}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{7}{3}$

Decimal Form:

$2.\overline{3}$

Mixed Number Form:

$2 \frac{1}{3}$

Step 8