Calculus Examples

$\int_{0}^{\frac{13 π}{6}} 8 \sin (x) + 4 \cos (x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{\frac{13 π}{6}} 8 \sin (x) d x + \int_{0}^{\frac{13 π}{6}} 4 \cos (x) d x$

Step 2

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int_{0}^{\frac{13 π}{6}} \sin (x) d x + \int_{0}^{\frac{13 π}{6}} 4 \cos (x) d x$

Step 3

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$8 (- \cos (x)]_{0}^{\frac{13 π}{6}}) + \int_{0}^{\frac{13 π}{6}} 4 \cos (x) d x$

Step 4

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

$8 (- \cos (x)]_{0}^{\frac{13 π}{6}}) + 4 \int_{0}^{\frac{13 π}{6}} \cos (x) d x$

Step 5

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$8 (- \cos (x)]_{0}^{\frac{13 π}{6}}) + 4 (\sin (x)]_{0}^{\frac{13 π}{6}})$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Substitute and simplify.

Tap for more steps...

Step 6.1.1

Evaluate $- \cos (x)$ at $\frac{13 π}{6}$ and at $0$ .

$8 ((- \cos (\frac{13 π}{6})) + \cos (0)) + 4 (\sin (x)]_{0}^{\frac{13 π}{6}})$

Step 6.1.2

Evaluate $\sin (x)$ at $\frac{13 π}{6}$ and at $0$ .

$8 (- \cos (\frac{13 π}{6}) + \cos (0)) + 4 (\sin (\frac{13 π}{6}) - \sin (0))$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

The exact value of $\cos (0)$ is $1$ .

$8 (- \cos (\frac{13 π}{6}) + 1) + 4 (\sin (\frac{13 π}{6}) - \sin (0))$

Step 6.2.2

The exact value of $\sin (0)$ is $0$ .

$8 (- \cos (\frac{13 π}{6}) + 1) + 4 (\sin (\frac{13 π}{6}) - 0)$

Step 6.2.3

Multiply $- 1$ by $0$ .

$8 (- \cos (\frac{13 π}{6}) + 1) + 4 (\sin (\frac{13 π}{6}) + 0)$

Step 6.2.4

Add $\sin (\frac{13 π}{6})$ and $0$ .

$8 (- \cos (\frac{13 π}{6}) + 1) + 4 \sin (\frac{13 π}{6})$

Step 6.3

Simplify.

Tap for more steps...

Step 6.3.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$8 (- \cos (\frac{π}{6}) + 1) + 4 \sin (\frac{13 π}{6})$

Step 6.3.2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$8 (- \frac{\sqrt{3}}{2} + 1) + 4 \sin (\frac{13 π}{6})$

Step 6.3.3

Apply the distributive property.

$8 (- \frac{\sqrt{3}}{2}) + 8 \cdot 1 + 4 \sin (\frac{13 π}{6})$

Step 6.3.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.4.1

Move the leading negative in $- \frac{\sqrt{3}}{2}$ into the numerator.

$8 \frac{- \sqrt{3}}{2} + 8 \cdot 1 + 4 \sin (\frac{13 π}{6})$

Step 6.3.4.2

Factor $2$ out of $8$ .

$2 (4) \frac{- \sqrt{3}}{2} + 8 \cdot 1 + 4 \sin (\frac{13 π}{6})$

Step 6.3.4.3

Cancel the common factor.

$2 \cdot 4 \frac{- \sqrt{3}}{2} + 8 \cdot 1 + 4 \sin (\frac{13 π}{6})$

Step 6.3.4.4

Rewrite the expression.

$4 (- \sqrt{3}) + 8 \cdot 1 + 4 \sin (\frac{13 π}{6})$

Step 6.3.5

Multiply $- 1$ by $4$ .

$- 4 \sqrt{3} + 8 \cdot 1 + 4 \sin (\frac{13 π}{6})$

Step 6.3.6

Multiply $8$ by $1$ .

$- 4 \sqrt{3} + 8 + 4 \sin (\frac{13 π}{6})$

Step 6.3.7

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- 4 \sqrt{3} + 8 + 4 \sin (\frac{π}{6})$

Step 6.3.8

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$- 4 \sqrt{3} + 8 + 4 (\frac{1}{2})$

Step 6.3.9

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.3.9.1

Factor $2$ out of $4$ .

$- 4 \sqrt{3} + 8 + 2 (2) \frac{1}{2}$

Step 6.3.9.2

Cancel the common factor.

$- 4 \sqrt{3} + 8 + 2 \cdot 2 \frac{1}{2}$

Step 6.3.9.3

Rewrite the expression.

$- 4 \sqrt{3} + 8 + 2$

Step 6.3.10

Add $8$ and $2$ .

$- 4 \sqrt{3} + 10$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- 4 \sqrt{3} + 10$

Decimal Form:

$3.07179676 \dots$