Calculus Examples

$\int_{0}^{1} (3 x - 2 \sqrt{x}) d x$

Step 1

Remove parentheses.

$\int_{0}^{1} 3 x - 2 \sqrt{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{1} 3 x d x + \int_{0}^{1} - 2 \sqrt{x} d x$

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{0}^{1} x d x + \int_{0}^{1} - 2 \sqrt{x} d x$

Step 4

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

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

Step 5

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

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

Step 6

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

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

Step 9.2

Substitute and simplify.

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.2.3

Simplify.

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Step 9.2.3.1

One to any power is one.

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

Step 9.2.3.2

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

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

Step 9.2.3.3

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

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Step 9.2.3.3.1

Factor $2$ out of $0$ .

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

Step 9.2.3.3.2

Cancel the common factors.

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Step 9.2.3.3.2.1

Factor $2$ out of $2$ .

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

Step 9.2.3.3.2.2

Cancel the common factor.

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

Step 9.2.3.3.2.3

Rewrite the expression.

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

Step 9.2.3.3.2.4

Divide $0$ by $1$ .

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

Step 9.2.3.4

Multiply $- 1$ by $0$ .

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

Step 9.2.3.5

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

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

Step 9.2.3.6

Combine $3$ and $\frac{1}{2}$ .

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

Step 9.2.3.7

One to any power is one.

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

Step 9.2.3.8

Multiply $2$ by $1$ .

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

Step 9.2.3.9

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

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

Step 9.2.3.10

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

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

Step 9.2.3.11

Cancel the common factor of $2$ .

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Step 9.2.3.11.1

Cancel the common factor.

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

Step 9.2.3.11.2

Rewrite the expression.

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

Step 9.2.3.12

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

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

Step 9.2.3.13

Multiply $2$ by $0$ .

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

Step 9.2.3.14

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

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Step 9.2.3.14.1

Factor $3$ out of $0$ .

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

Step 9.2.3.14.2

Cancel the common factors.

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Step 9.2.3.14.2.1

Factor $3$ out of $3$ .

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

Step 9.2.3.14.2.2

Cancel the common factor.

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

Step 9.2.3.14.2.3

Rewrite the expression.

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

Step 9.2.3.14.2.4

Divide $0$ by $1$ .

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

Step 9.2.3.15

Multiply $- 1$ by $0$ .

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

Step 9.2.3.16

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

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

Step 9.2.3.17

Combine $- 2$ and $\frac{2}{3}$ .

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

Step 9.2.3.18

Multiply $- 2$ by $2$ .

$\frac{3}{2} + \frac{- 4}{3}$

Step 9.2.3.19

Move the negative in front of the fraction.

$\frac{3}{2} - \frac{4}{3}$

Step 9.2.3.20

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

$\frac{3}{2} \cdot \frac{3}{3} - \frac{4}{3}$

Step 9.2.3.21

To write $- \frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} \cdot \frac{3}{3} - \frac{4}{3} \cdot \frac{2}{2}$

Step 9.2.3.22

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

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Step 9.2.3.22.1

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

$\frac{3 \cdot 3}{2 \cdot 3} - \frac{4}{3} \cdot \frac{2}{2}$

Step 9.2.3.22.2

Multiply $2$ by $3$ .

$\frac{3 \cdot 3}{6} - \frac{4}{3} \cdot \frac{2}{2}$

Step 9.2.3.22.3

Multiply $\frac{4}{3}$ by $\frac{2}{2}$ .

$\frac{3 \cdot 3}{6} - \frac{4 \cdot 2}{3 \cdot 2}$

Step 9.2.3.22.4

Multiply $3$ by $2$ .

$\frac{3 \cdot 3}{6} - \frac{4 \cdot 2}{6}$

Step 9.2.3.23

Combine the numerators over the common denominator.

$\frac{3 \cdot 3 - 4 \cdot 2}{6}$

Step 9.2.3.24

Simplify the numerator.

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Step 9.2.3.24.1

Multiply $3$ by $3$ .

$\frac{9 - 4 \cdot 2}{6}$

Step 9.2.3.24.2

Multiply $- 4$ by $2$ .

$\frac{9 - 8}{6}$

Step 9.2.3.24.3

Subtract $8$ from $9$ .

$\frac{1}{6}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$

Step 11