Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

$- (- \cos (\frac{π}{2}) + \cos (- \frac{π}{2}))$

Step 3.2

Simplify.

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Step 3.2.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- (- 0 + \cos (- \frac{π}{2}))$

Step 3.2.2

Multiply $- 1$ by $0$ .

$- (0 + \cos (- \frac{π}{2}))$

Step 3.2.3

Add $0$ and $\cos (- \frac{π}{2})$ .

$- \cos (- \frac{π}{2})$

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 $2 π$ .

$- \cos (\frac{3 π}{2})$

Step 3.3.2

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

$- \cos (\frac{π}{2})$

Step 3.3.3

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- 0$

Step 3.3.4

Multiply $- 1$ by $0$ .

$0$