Calculus Examples

$\int_{1}^{3} 2 x^{4} - 3 x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{3} 2 x^{4} d x + \int_{1}^{3} - 3 x^{2} d x$

Step 2

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

$2 \int_{1}^{3} x^{4} d x + \int_{1}^{3} - 3 x^{2} d x$

Step 3

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$2 (\frac{1}{5} x^{5}]_{1}^{3}) + \int_{1}^{3} - 3 x^{2} d x$

Step 4

Combine $\frac{1}{5}$ and $x^{5}$ .

$2 (\frac{x^{5}}{5}]_{1}^{3}) + \int_{1}^{3} - 3 x^{2} d x$

Step 5

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

$2 (\frac{x^{5}}{5}]_{1}^{3}) - 3 \int_{1}^{3} x^{2} d x$

Step 6

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

$2 (\frac{x^{5}}{5}]_{1}^{3}) - 3 (\frac{1}{3} x^{3}]_{1}^{3})$

Step 7

Simplify the answer.

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Step 7.1

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

$2 (\frac{x^{5}}{5}]_{1}^{3}) - 3 (\frac{x^{3}}{3}]_{1}^{3})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{x^{5}}{5}$ at $3$ and at $1$ .

$2 ((\frac{3^{5}}{5}) - \frac{1^{5}}{5}) - 3 (\frac{x^{3}}{3}]_{1}^{3})$

Step 7.2.2

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

$2 (\frac{3^{5}}{5} - \frac{1^{5}}{5}) - 3 (\frac{3^{3}}{3} - \frac{1^{3}}{3})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Raise $3$ to the power of $5$ .

$2 (\frac{243}{5} - \frac{1^{5}}{5}) - 3 (\frac{3^{3}}{3} - \frac{1^{3}}{3})$

Step 7.2.3.2

One to any power is one.

$2 (\frac{243}{5} - \frac{1}{5}) - 3 (\frac{3^{3}}{3} - \frac{1^{3}}{3})$

Step 7.2.3.3

Combine the numerators over the common denominator.

$2 \frac{243 - 1}{5} - 3 (\frac{3^{3}}{3} - \frac{1^{3}}{3})$

Step 7.2.3.4

Subtract $1$ from $243$ .

$2 (\frac{242}{5}) - 3 (\frac{3^{3}}{3} - \frac{1^{3}}{3})$

Step 7.2.3.5

Combine $2$ and $\frac{242}{5}$ .

$\frac{2 \cdot 242}{5} - 3 (\frac{3^{3}}{3} - \frac{1^{3}}{3})$

Step 7.2.3.6

Multiply $2$ by $242$ .

$\frac{484}{5} - 3 (\frac{3^{3}}{3} - \frac{1^{3}}{3})$

Step 7.2.3.7

Raise $3$ to the power of $3$ .

$\frac{484}{5} - 3 (\frac{27}{3} - \frac{1^{3}}{3})$

Step 7.2.3.8

Cancel the common factor of $27$ and $3$ .

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Step 7.2.3.8.1

Factor $3$ out of $27$ .

$\frac{484}{5} - 3 (\frac{3 \cdot 9}{3} - \frac{1^{3}}{3})$

Step 7.2.3.8.2

Cancel the common factors.

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Step 7.2.3.8.2.1

Factor $3$ out of $3$ .

$\frac{484}{5} - 3 (\frac{3 \cdot 9}{3 (1)} - \frac{1^{3}}{3})$

Step 7.2.3.8.2.2

Cancel the common factor.

$\frac{484}{5} - 3 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{1^{3}}{3})$

Step 7.2.3.8.2.3

Rewrite the expression.

$\frac{484}{5} - 3 (\frac{9}{1} - \frac{1^{3}}{3})$

Step 7.2.3.8.2.4

Divide $9$ by $1$ .

$\frac{484}{5} - 3 (9 - \frac{1^{3}}{3})$

Step 7.2.3.9

One to any power is one.

$\frac{484}{5} - 3 (9 - \frac{1}{3})$

Step 7.2.3.10

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

$\frac{484}{5} - 3 (9 \cdot \frac{3}{3} - \frac{1}{3})$

Step 7.2.3.11

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

$\frac{484}{5} - 3 (\frac{9 \cdot 3}{3} - \frac{1}{3})$

Step 7.2.3.12

Combine the numerators over the common denominator.

$\frac{484}{5} - 3 \frac{9 \cdot 3 - 1}{3}$

Step 7.2.3.13

Simplify the numerator.

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Step 7.2.3.13.1

Multiply $9$ by $3$ .

$\frac{484}{5} - 3 \frac{27 - 1}{3}$

Step 7.2.3.13.2

Subtract $1$ from $27$ .

$\frac{484}{5} - 3 (\frac{26}{3})$

Step 7.2.3.14

Combine $- 3$ and $\frac{26}{3}$ .

$\frac{484}{5} + \frac{- 3 \cdot 26}{3}$

Step 7.2.3.15

Multiply $- 3$ by $26$ .

$\frac{484}{5} + \frac{- 78}{3}$

Step 7.2.3.16

Cancel the common factor of $- 78$ and $3$ .

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Step 7.2.3.16.1

Factor $3$ out of $- 78$ .

$\frac{484}{5} + \frac{3 \cdot - 26}{3}$

Step 7.2.3.16.2

Cancel the common factors.

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Step 7.2.3.16.2.1

Factor $3$ out of $3$ .

$\frac{484}{5} + \frac{3 \cdot - 26}{3 (1)}$

Step 7.2.3.16.2.2

Cancel the common factor.

$\frac{484}{5} + \frac{3 \cdot - 26}{3 \cdot 1}$

Step 7.2.3.16.2.3

Rewrite the expression.

$\frac{484}{5} + \frac{- 26}{1}$

Step 7.2.3.16.2.4

Divide $- 26$ by $1$ .

$\frac{484}{5} - 26$

Step 7.2.3.17

To write $- 26$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{484}{5} - 26 \cdot \frac{5}{5}$

Step 7.2.3.18

Combine $- 26$ and $\frac{5}{5}$ .

$\frac{484}{5} + \frac{- 26 \cdot 5}{5}$

Step 7.2.3.19

Combine the numerators over the common denominator.

$\frac{484 - 26 \cdot 5}{5}$

Step 7.2.3.20

Simplify the numerator.

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Step 7.2.3.20.1

Multiply $- 26$ by $5$ .

$\frac{484 - 130}{5}$

Step 7.2.3.20.2

Subtract $130$ from $484$ .

$\frac{354}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{354}{5}$

Decimal Form:

$70.8$

Mixed Number Form:

$70 \frac{4}{5}$

Step 9