Calculus Examples

$\int_{0}^{3} x^{3} - 4 x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{3} x^{3} d x + \int_{0}^{3} - 4 x d x$

Step 2

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 3

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

$\frac{1}{4} x^{4}]_{0}^{3} - 4 \int_{0}^{3} x d x$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{1}{4} x^{4}$ at $3$ and at $0$ .

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

Step 5.2.2

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

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

Step 5.2.3

Simplify.

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Step 5.2.3.1

Raise $3$ to the power of $4$ .

$\frac{1}{4} \cdot 81 - \frac{1}{4} \cdot 0^{4} - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.2

Combine $\frac{1}{4}$ and $81$ .

$\frac{81}{4} - \frac{1}{4} \cdot 0^{4} - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.3

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

$\frac{81}{4} - \frac{1}{4} \cdot 0 - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.4

Multiply $0$ by $- 1$ .

$\frac{81}{4} + 0 (\frac{1}{4}) - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.5

Multiply $0$ by $\frac{1}{4}$ .

$\frac{81}{4} + 0 - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.6

Add $\frac{81}{4}$ and $0$ .

$\frac{81}{4} - 4 (\frac{3^{2}}{2} - \frac{0^{2}}{2})$

Step 5.2.3.7

Raise $3$ to the power of $2$ .

$\frac{81}{4} - 4 (\frac{9}{2} - \frac{0^{2}}{2})$

Step 5.2.3.8

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

$\frac{81}{4} - 4 (\frac{9}{2} - \frac{0}{2})$

Step 5.2.3.9

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

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Step 5.2.3.9.1

Factor $2$ out of $0$ .

$\frac{81}{4} - 4 (\frac{9}{2} - \frac{2 (0)}{2})$

Step 5.2.3.9.2

Cancel the common factors.

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Step 5.2.3.9.2.1

Factor $2$ out of $2$ .

$\frac{81}{4} - 4 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.3.9.2.2

Cancel the common factor.

$\frac{81}{4} - 4 (\frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.2.3.9.2.3

Rewrite the expression.

$\frac{81}{4} - 4 (\frac{9}{2} - \frac{0}{1})$

Step 5.2.3.9.2.4

Divide $0$ by $1$ .

$\frac{81}{4} - 4 (\frac{9}{2} - 0)$

Step 5.2.3.10

Multiply $- 1$ by $0$ .

$\frac{81}{4} - 4 (\frac{9}{2} + 0)$

Step 5.2.3.11

Add $\frac{9}{2}$ and $0$ .

$\frac{81}{4} - 4 (\frac{9}{2})$

Step 5.2.3.12

Combine $- 4$ and $\frac{9}{2}$ .

$\frac{81}{4} + \frac{- 4 \cdot 9}{2}$

Step 5.2.3.13

Multiply $- 4$ by $9$ .

$\frac{81}{4} + \frac{- 36}{2}$

Step 5.2.3.14

Cancel the common factor of $- 36$ and $2$ .

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Step 5.2.3.14.1

Factor $2$ out of $- 36$ .

$\frac{81}{4} + \frac{2 \cdot - 18}{2}$

Step 5.2.3.14.2

Cancel the common factors.

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Step 5.2.3.14.2.1

Factor $2$ out of $2$ .

$\frac{81}{4} + \frac{2 \cdot - 18}{2 (1)}$

Step 5.2.3.14.2.2

Cancel the common factor.

$\frac{81}{4} + \frac{2 \cdot - 18}{2 \cdot 1}$

Step 5.2.3.14.2.3

Rewrite the expression.

$\frac{81}{4} + \frac{- 18}{1}$

Step 5.2.3.14.2.4

Divide $- 18$ by $1$ .

$\frac{81}{4} - 18$

Step 5.2.3.15

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{81}{4} - 18 \cdot \frac{4}{4}$

Step 5.2.3.16

Combine $- 18$ and $\frac{4}{4}$ .

$\frac{81}{4} + \frac{- 18 \cdot 4}{4}$

Step 5.2.3.17

Combine the numerators over the common denominator.

$\frac{81 - 18 \cdot 4}{4}$

Step 5.2.3.18

Simplify the numerator.

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Step 5.2.3.18.1

Multiply $- 18$ by $4$ .

$\frac{81 - 72}{4}$

Step 5.2.3.18.2

Subtract $72$ from $81$ .

$\frac{9}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{4}$

Decimal Form:

$2.25$

Mixed Number Form:

$2 \frac{1}{4}$

Step 7