Calculus Examples

$\int_{- \frac{π}{2}}^{π} 11 \cos (x) d x$

Step 1

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$11 \int_{- \frac{π}{2}}^{π} \cos (x) d x$

Step 2

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

$11 (\sin (x)]_{- \frac{π}{2}}^{π})$

Step 3

Evaluate $\sin (x)$ at $π$ and at $- \frac{π}{2}$ .

$11 (\sin (π) - \sin (- \frac{π}{2}))$

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$11 (\sin (0) - \sin (- \frac{π}{2}))$

Step 4.1.2

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

$11 (0 - \sin (- \frac{π}{2}))$

Step 4.1.3

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

$11 (0 - \sin (\frac{3 π}{2}))$

Step 4.1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$11 (0 - - \sin (\frac{π}{2}))$

Step 4.1.5

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

$11 (0 - (- 1 \cdot 1))$

Step 4.1.6

Multiply $- (- 1 \cdot 1)$ .

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Step 4.1.6.1

Multiply $- 1$ by $1$ .

$11 (0 - - 1)$

Step 4.1.6.2

Multiply $- 1$ by $- 1$ .

$11 (0 + 1)$

Step 4.2

Add $0$ and $1$ .

$11 \cdot 1$

Step 4.3

Multiply $11$ by $1$ .

$11$