Calculus Examples

$\int_{1}^{3} 8 - 2 x d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{3} 8 d x + \int_{1}^{3} - 2 x d x$

Step 2

Apply the constant rule.

$8 x]_{1}^{3} + \int_{1}^{3} - 2 x d x$

Step 3

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

$8 x]_{1}^{3} - 2 \int_{1}^{3} x d x$

Step 4

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

$8 x]_{1}^{3} - 2 (\frac{1}{2} x^{2}]_{1}^{3})$

Step 5

Simplify the answer.

Tap for more steps...

Step 5.1

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

$8 x]_{1}^{3} - 2 (\frac{x^{2}}{2}]_{1}^{3})$

Step 5.2

Substitute and simplify.

Tap for more steps...

Step 5.2.1

Evaluate $8 x$ at $3$ and at $1$ .

$(8 \cdot 3) - 8 \cdot 1 - 2 (\frac{x^{2}}{2}]_{1}^{3})$

Step 5.2.2

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

$8 \cdot 3 - 8 \cdot 1 - 2 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3

Simplify.

Tap for more steps...

Step 5.2.3.1

Multiply $8$ by $3$ .

$24 - 8 \cdot 1 - 2 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3.2

Multiply $- 8$ by $1$ .

$24 - 8 - 2 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3.3

Subtract $8$ from $24$ .

$16 - 2 (\frac{3^{2}}{2} - \frac{1^{2}}{2})$

Step 5.2.3.4

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

$16 - 2 (\frac{9}{2} - \frac{1^{2}}{2})$

Step 5.2.3.5

One to any power is one.

$16 - 2 (\frac{9}{2} - \frac{1}{2})$

Step 5.2.3.6

Combine the numerators over the common denominator.

$16 - 2 \frac{9 - 1}{2}$

Step 5.2.3.7

Subtract $1$ from $9$ .

$16 - 2 (\frac{8}{2})$

Step 5.2.3.8

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

Tap for more steps...

Step 5.2.3.8.1

Factor $2$ out of $8$ .

$16 - 2 \frac{2 \cdot 4}{2}$

Step 5.2.3.8.2

Cancel the common factors.

Tap for more steps...

Step 5.2.3.8.2.1

Factor $2$ out of $2$ .

$16 - 2 \frac{2 \cdot 4}{2 (1)}$

Step 5.2.3.8.2.2

Cancel the common factor.

$16 - 2 \frac{2 \cdot 4}{2 \cdot 1}$

Step 5.2.3.8.2.3

Rewrite the expression.

$16 - 2 (\frac{4}{1})$

Step 5.2.3.8.2.4

Divide $4$ by $1$ .

$16 - 2 \cdot 4$

Step 5.2.3.9

Multiply $- 2$ by $4$ .

$16 - 8$

Step 5.2.3.10

Subtract $8$ from $16$ .

$8$

Step 6