Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{1}{2}$ and $x^{2}$ .

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

Step 6.2

Substitute and simplify.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

One to any power is one.

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

Step 6.2.3.2

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

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

Step 6.2.3.3

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

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

Step 6.2.3.4

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

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

Step 6.2.3.5

Cancel the common factor of $2$ .

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Step 6.2.3.5.1

Cancel the common factor.

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

Step 6.2.3.5.2

Rewrite the expression.

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

Step 6.2.3.6

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

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

Step 6.2.3.7

Multiply $0$ by $- 1$ .

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

Step 6.2.3.8

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

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

Step 6.2.3.9

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

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

Step 6.2.3.10

One to any power is one.

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

Step 6.2.3.11

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

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

Step 6.2.3.12

Cancel the common factor of $0$ and $2$ .

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Step 6.2.3.12.1

Factor $2$ out of $0$ .

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

Step 6.2.3.12.2

Cancel the common factors.

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Step 6.2.3.12.2.1

Factor $2$ out of $2$ .

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

Step 6.2.3.12.2.2

Cancel the common factor.

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

Step 6.2.3.12.2.3

Rewrite the expression.

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

Step 6.2.3.12.2.4

Divide $0$ by $1$ .

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

Step 6.2.3.13

Multiply $- 1$ by $0$ .

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

Step 6.2.3.14

Add $\frac{1}{2}$ and $0$ .

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

Step 6.2.3.15

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.2.3.16

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6.2.3.17

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.3.17.1

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

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

Step 6.2.3.17.2

Multiply $3$ by $2$ .

$\frac{2 \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 6.2.3.17.3

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

$\frac{2 \cdot 2}{6} - \frac{3}{2 \cdot 3}$

Step 6.2.3.17.4

Multiply $2$ by $3$ .

$\frac{2 \cdot 2}{6} - \frac{3}{6}$

Step 6.2.3.18

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 - 3}{6}$

Step 6.2.3.19

Simplify the numerator.

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Step 6.2.3.19.1

Multiply $2$ by $2$ .

$\frac{4 - 3}{6}$

Step 6.2.3.19.2

Subtract $3$ from $4$ .

$\frac{1}{6}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$

Step 8