Calculus Examples

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

Step 1

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

$\frac{1}{4} x^{4}]_{- 1}^{2}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{4} x^{4}$ at $2$ and at $- 1$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Raise $2$ to the power of $4$ .

$\frac{1}{4} \cdot 16 - \frac{1}{4} {(- 1)}^{4}$

Step 2.2.2

Combine $\frac{1}{4}$ and $16$ .

$\frac{16}{4} - \frac{1}{4} {(- 1)}^{4}$

Step 2.2.3

Cancel the common factor of $16$ and $4$ .

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Step 2.2.3.1

Factor $4$ out of $16$ .

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

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $4$ out of $4$ .

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

Step 2.2.3.2.2

Cancel the common factor.

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

Step 2.2.3.2.3

Rewrite the expression.

$\frac{4}{1} - \frac{1}{4} {(- 1)}^{4}$

Step 2.2.3.2.4

Divide $4$ by $1$ .

$4 - \frac{1}{4} {(- 1)}^{4}$

Step 2.2.4

Raise $- 1$ to the power of $4$ .

$4 - \frac{1}{4} \cdot 1$

Step 2.2.5

Multiply $- 1$ by $1$ .

$4 - \frac{1}{4}$

Step 2.2.6

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

$4 \cdot \frac{4}{4} - \frac{1}{4}$

Step 2.2.7

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

$\frac{4 \cdot 4}{4} - \frac{1}{4}$

Step 2.2.8

Combine the numerators over the common denominator.

$\frac{4 \cdot 4 - 1}{4}$

Step 2.2.9

Simplify the numerator.

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Step 2.2.9.1

Multiply $4$ by $4$ .

$\frac{16 - 1}{4}$

Step 2.2.9.2

Subtract $1$ from $16$ .

$\frac{15}{4}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{15}{4}$

Decimal Form:

$3.75$

Mixed Number Form:

$3 \frac{3}{4}$

Step 4