Calculus Examples

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

Step 1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 2

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

$- (\sin (x)]_{- \frac{π}{2}}^{\frac{π}{2}})$

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

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

$- (1 - \sin (- \frac{π}{2}))$

Step 3.3

Simplify.

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Step 3.3.1

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

$- (1 - \sin (\frac{3 π}{2}))$

Step 3.3.2

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.

$- (1 - - \sin (\frac{π}{2}))$

Step 3.3.3

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

$- (1 - (- 1 \cdot 1))$

Step 3.3.4

Multiply $- 1$ by $1$ .

$- (1 - - 1)$

Step 3.3.5

Multiply $- 1$ by $- 1$ .

$- (1 + 1)$

Step 3.3.6

Add $1$ and $1$ .

$- 1 \cdot 2$

Step 3.3.7

Multiply $- 1$ by $2$ .

$- 2$