Calculus Examples

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Remove parentheses.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the answer.

Tap for more steps...

Step 7.1

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

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

Step 7.2

Substitute and simplify.

Tap for more steps...

Step 7.2.1

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

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

Step 7.2.2

Evaluate $\frac{x^{2}}{2}$ at $3$ and at $1$ .

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

Step 7.2.3

Simplify.

Tap for more steps...

Step 7.2.3.1

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

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

Step 7.2.3.2

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

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

Step 7.2.3.3

One to any power is one.

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

Step 7.2.3.4

Multiply $- 1$ by $1$ .

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

Step 7.2.3.5

Combine the numerators over the common denominator.

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

Step 7.2.3.6

Subtract $1$ from $81$ .

$\frac{80}{4} - 4 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 7.2.3.7

Cancel the common factor of $80$ and $4$ .

Tap for more steps...

Step 7.2.3.7.1

Factor $4$ out of $80$ .

$\frac{4 \cdot 20}{4} - 4 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 7.2.3.7.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.7.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 20}{4 (1)} - 4 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 7.2.3.7.2.2

Cancel the common factor.

$\frac{4 \cdot 20}{4 \cdot 1} - 4 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 7.2.3.7.2.3

Rewrite the expression.

$\frac{20}{1} - 4 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 7.2.3.7.2.4

Divide $20$ by $1$ .

$20 - 4 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 7.2.3.8

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

$20 - 4 (\frac{9}{2} - \frac{1^{2}}{2})$

Step 7.2.3.9

One to any power is one.

$20 - 4 (\frac{9}{2} - \frac{1}{2})$

Step 7.2.3.10

Combine the numerators over the common denominator.

$20 - 4 \frac{9 - 1}{2}$

Step 7.2.3.11

Subtract $1$ from $9$ .

$20 - 4 (\frac{8}{2})$

Step 7.2.3.12

Cancel the common factor of $8$ and $2$ .

Tap for more steps...

Step 7.2.3.12.1

Factor $2$ out of $8$ .

$20 - 4 \frac{2 \cdot 4}{2}$

Step 7.2.3.12.2

Cancel the common factors.

Tap for more steps...

Step 7.2.3.12.2.1

Factor $2$ out of $2$ .

$20 - 4 \frac{2 \cdot 4}{2 (1)}$

Step 7.2.3.12.2.2

Cancel the common factor.

$20 - 4 \frac{2 \cdot 4}{2 \cdot 1}$

Step 7.2.3.12.2.3

Rewrite the expression.

$20 - 4 (\frac{4}{1})$

Step 7.2.3.12.2.4

Divide $4$ by $1$ .

$20 - 4 \cdot 4$

Step 7.2.3.13

Multiply $- 4$ by $4$ .

$20 - 16$

Step 7.2.3.14

Subtract $16$ from $20$ .

$4$

Step 8