Calculus Examples

$\int_{0}^{1} 7 x^{2} - 4 x + 9 d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{1} 7 x^{2} d x + \int_{0}^{1} - 4 x d x + \int_{0}^{1} 9 d x$

Step 2

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int_{0}^{1} x^{2} d x + \int_{0}^{1} - 4 x d x + \int_{0}^{1} 9 d x$

Step 3

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

$7 (\frac{1}{3} x^{3}]_{0}^{1}) + \int_{0}^{1} - 4 x d x + \int_{0}^{1} 9 d x$

Step 4

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

$7 (\frac{x^{3}}{3}]_{0}^{1}) + \int_{0}^{1} - 4 x d x + \int_{0}^{1} 9 d x$

Step 5

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$7 (\frac{x^{3}}{3}]_{0}^{1}) - 4 \int_{0}^{1} x d x + \int_{0}^{1} 9 d x$

Step 6

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

$7 (\frac{x^{3}}{3}]_{0}^{1}) - 4 (\frac{1}{2} x^{2}]_{0}^{1}) + \int_{0}^{1} 9 d x$

Step 7

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

$7 (\frac{x^{3}}{3}]_{0}^{1}) - 4 (\frac{x^{2}}{2}]_{0}^{1}) + \int_{0}^{1} 9 d x$

Step 8

Apply the constant rule.

$7 (\frac{x^{3}}{3}]_{0}^{1}) - 4 (\frac{x^{2}}{2}]_{0}^{1}) + 9 x]_{0}^{1}$

Step 9

Substitute and simplify.

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Step 9.1

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

$7 ((\frac{1^{3}}{3}) - \frac{0^{3}}{3}) - 4 (\frac{x^{2}}{2}]_{0}^{1}) + 9 x]_{0}^{1}$

Step 9.2

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

$7 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + 9 x]_{0}^{1}$

Step 9.3

Evaluate $9 x$ at $1$ and at $0$ .

$7 (\frac{1^{3}}{3} - \frac{0^{3}}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4

Simplify.

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Step 9.4.1

One to any power is one.

$7 (\frac{1}{3} - \frac{0^{3}}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.2

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

$7 (\frac{1}{3} - \frac{0}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.3

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

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Step 9.4.3.1

Factor $3$ out of $0$ .

$7 (\frac{1}{3} - \frac{3 (0)}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.3.2

Cancel the common factors.

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Step 9.4.3.2.1

Factor $3$ out of $3$ .

$7 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.3.2.2

Cancel the common factor.

$7 (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.3.2.3

Rewrite the expression.

$7 (\frac{1}{3} - \frac{0}{1}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.3.2.4

Divide $0$ by $1$ .

$7 (\frac{1}{3} - 0) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.4

Multiply $- 1$ by $0$ .

$7 (\frac{1}{3} + 0) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.5

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

$7 (\frac{1}{3}) - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.6

Combine $7$ and $\frac{1}{3}$ .

$\frac{7}{3} - 4 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.7

One to any power is one.

$\frac{7}{3} - 4 (\frac{1}{2} - \frac{0^{2}}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.8

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

$\frac{7}{3} - 4 (\frac{1}{2} - \frac{0}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.9

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

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Step 9.4.9.1

Factor $2$ out of $0$ .

$\frac{7}{3} - 4 (\frac{1}{2} - \frac{2 (0)}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.9.2

Cancel the common factors.

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Step 9.4.9.2.1

Factor $2$ out of $2$ .

$\frac{7}{3} - 4 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.9.2.2

Cancel the common factor.

$\frac{7}{3} - 4 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.9.2.3

Rewrite the expression.

$\frac{7}{3} - 4 (\frac{1}{2} - \frac{0}{1}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.9.2.4

Divide $0$ by $1$ .

$\frac{7}{3} - 4 (\frac{1}{2} - 0) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.10

Multiply $- 1$ by $0$ .

$\frac{7}{3} - 4 (\frac{1}{2} + 0) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.11

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

$\frac{7}{3} - 4 (\frac{1}{2}) + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.12

Combine $- 4$ and $\frac{1}{2}$ .

$\frac{7}{3} + \frac{- 4}{2} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.13

Cancel the common factor of $- 4$ and $2$ .

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Step 9.4.13.1

Factor $2$ out of $- 4$ .

$\frac{7}{3} + \frac{2 \cdot - 2}{2} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.13.2

Cancel the common factors.

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Step 9.4.13.2.1

Factor $2$ out of $2$ .

$\frac{7}{3} + \frac{2 \cdot - 2}{2 (1)} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.13.2.2

Cancel the common factor.

$\frac{7}{3} + \frac{2 \cdot - 2}{2 \cdot 1} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.13.2.3

Rewrite the expression.

$\frac{7}{3} + \frac{- 2}{1} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.13.2.4

Divide $- 2$ by $1$ .

$\frac{7}{3} - 2 + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.14

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

$\frac{7}{3} - 2 \cdot \frac{3}{3} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.15

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

$\frac{7}{3} + \frac{- 2 \cdot 3}{3} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.16

Combine the numerators over the common denominator.

$\frac{7 - 2 \cdot 3}{3} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.17

Simplify the numerator.

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Step 9.4.17.1

Multiply $- 2$ by $3$ .

$\frac{7 - 6}{3} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.17.2

Subtract $6$ from $7$ .

$\frac{1}{3} + (9 \cdot 1) - 9 \cdot 0$

Step 9.4.18

Multiply $9$ by $1$ .

$\frac{1}{3} + 9 - 9 \cdot 0$

Step 9.4.19

Multiply $- 9$ by $0$ .

$\frac{1}{3} + 9 + 0$

Step 9.4.20

Add $9$ and $0$ .

$\frac{1}{3} + 9$

Step 9.4.21

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{3} + 9 \cdot \frac{3}{3}$

Step 9.4.22

Combine $9$ and $\frac{3}{3}$ .

$\frac{1}{3} + \frac{9 \cdot 3}{3}$

Step 9.4.23

Combine the numerators over the common denominator.

$\frac{1 + 9 \cdot 3}{3}$

Step 9.4.24

Simplify the numerator.

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Step 9.4.24.1

Multiply $9$ by $3$ .

$\frac{1 + 27}{3}$

Step 9.4.24.2

Add $1$ and $27$ .

$\frac{28}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{28}{3}$

Decimal Form:

$9.\overline{3}$

Mixed Number Form:

$9 \frac{1}{3}$

Step 11