Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 1.4

Move $x$ .

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

Step 1.5

Raise $x$ to the power of $1$ .

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

Step 1.6

Raise $x$ to the power of $1$ .

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

Step 1.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.8

Add $1$ and $1$ .

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

Step 1.9

Multiply $2$ by $2$ .

$\int_{0}^{1} 2 x^{2} + 4 x - 1 x - 1 \cdot 2 d x$

Step 1.10

Multiply $- 1$ by $2$ .

$\int_{0}^{1} 2 x^{2} + 4 x - 1 x - 2 d x$

Step 1.11

Subtract $1 x$ from $4 x$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

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

Step 10

Substitute and simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

Evaluate $- 2 x$ at $1$ and at $0$ .

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

Step 10.4

Simplify.

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Step 10.4.1

One to any power is one.

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

Step 10.4.2

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

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

Step 10.4.3

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

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Step 10.4.3.1

Factor $3$ out of $0$ .

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

Step 10.4.3.2

Cancel the common factors.

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Step 10.4.3.2.1

Factor $3$ out of $3$ .

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

Step 10.4.3.2.2

Cancel the common factor.

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

Step 10.4.3.2.3

Rewrite the expression.

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

Step 10.4.3.2.4

Divide $0$ by $1$ .

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

Step 10.4.4

Multiply $- 1$ by $0$ .

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

Step 10.4.5

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

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

Step 10.4.6

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

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

Step 10.4.7

One to any power is one.

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

Step 10.4.8

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

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

Step 10.4.9

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

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Step 10.4.9.1

Factor $2$ out of $0$ .

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

Step 10.4.9.2

Cancel the common factors.

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Step 10.4.9.2.1

Factor $2$ out of $2$ .

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

Step 10.4.9.2.2

Cancel the common factor.

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

Step 10.4.9.2.3

Rewrite the expression.

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

Step 10.4.9.2.4

Divide $0$ by $1$ .

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

Step 10.4.10

Multiply $- 1$ by $0$ .

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

Step 10.4.11

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

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

Step 10.4.12

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

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

Step 10.4.13

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

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

Step 10.4.14

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

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

Step 10.4.15

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

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Step 10.4.15.1

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

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

Step 10.4.15.2

Multiply $3$ by $2$ .

$\frac{2 \cdot 2}{6} + \frac{3}{2} \cdot \frac{3}{3} + (- 2 \cdot 1) + 2 \cdot 0$

Step 10.4.15.3

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

$\frac{2 \cdot 2}{6} + \frac{3 \cdot 3}{2 \cdot 3} + (- 2 \cdot 1) + 2 \cdot 0$

Step 10.4.15.4

Multiply $2$ by $3$ .

$\frac{2 \cdot 2}{6} + \frac{3 \cdot 3}{6} + (- 2 \cdot 1) + 2 \cdot 0$

Step 10.4.16

Combine the numerators over the common denominator.

$\frac{2 \cdot 2 + 3 \cdot 3}{6} + (- 2 \cdot 1) + 2 \cdot 0$

Step 10.4.17

Simplify the numerator.

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Step 10.4.17.1

Multiply $2$ by $2$ .

$\frac{4 + 3 \cdot 3}{6} + (- 2 \cdot 1) + 2 \cdot 0$

Step 10.4.17.2

Multiply $3$ by $3$ .

$\frac{4 + 9}{6} + (- 2 \cdot 1) + 2 \cdot 0$

Step 10.4.17.3

Add $4$ and $9$ .

$\frac{13}{6} + (- 2 \cdot 1) + 2 \cdot 0$

Step 10.4.18

Multiply $- 2$ by $1$ .

$\frac{13}{6} - 2 + 2 \cdot 0$

Step 10.4.19

Multiply $2$ by $0$ .

$\frac{13}{6} - 2 + 0$

Step 10.4.20

Add $- 2$ and $0$ .

$\frac{13}{6} - 2$

Step 10.4.21

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{13}{6} - 2 \cdot \frac{6}{6}$

Step 10.4.22

Combine $- 2$ and $\frac{6}{6}$ .

$\frac{13}{6} + \frac{- 2 \cdot 6}{6}$

Step 10.4.23

Combine the numerators over the common denominator.

$\frac{13 - 2 \cdot 6}{6}$

Step 10.4.24

Simplify the numerator.

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Step 10.4.24.1

Multiply $- 2$ by $6$ .

$\frac{13 - 12}{6}$

Step 10.4.24.2

Subtract $12$ from $13$ .

$\frac{1}{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$

Step 12