Calculus Examples

$\int_{0}^{1} (\sqrt{x} - x^{2}) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Substitute and simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

Simplify.

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Step 7.2.3.1

One to any power is one.

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

Step 7.2.3.2

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

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

Step 7.2.3.3

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

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

Step 7.2.3.4

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

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

Step 7.2.3.5

Cancel the common factor of $2$ .

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Step 7.2.3.5.1

Cancel the common factor.

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

Step 7.2.3.5.2

Rewrite the expression.

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

Step 7.2.3.6

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

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

Step 7.2.3.7

Multiply $0$ by $- 1$ .

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

Step 7.2.3.8

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

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

Step 7.2.3.9

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

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

Step 7.2.3.10

One to any power is one.

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

Step 7.2.3.11

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

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

Step 7.2.3.12

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

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Step 7.2.3.12.1

Factor $3$ out of $0$ .

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

Step 7.2.3.12.2

Cancel the common factors.

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Step 7.2.3.12.2.1

Factor $3$ out of $3$ .

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

Step 7.2.3.12.2.2

Cancel the common factor.

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

Step 7.2.3.12.2.3

Rewrite the expression.

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

Step 7.2.3.12.2.4

Divide $0$ by $1$ .

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

Step 7.2.3.13

Multiply $- 1$ by $0$ .

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

Step 7.2.3.14

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

$\frac{2}{3} - \frac{1}{3}$

Step 7.2.3.15

Combine the numerators over the common denominator.

$\frac{2 - 1}{3}$

Step 7.2.3.16

Subtract $1$ from $2$ .

$\frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$

Step 9