Calculus Examples

$\int_{- π}^{π} \sin (x) d x$

Step 1

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

$- \cos (x)]_{- π}^{π}$

Step 2

Evaluate $- \cos (x)$ at $π$ and at $- π$ .

$- \cos (π) + \cos (- π)$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$- - \cos (0) + \cos (- π)$

Step 3.1.2

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

$- (- 1 \cdot 1) + \cos (- π)$

Step 3.1.3

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

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Step 3.1.3.1

Multiply $- 1$ by $1$ .

$- - 1 + \cos (- π)$

Step 3.1.3.2

Multiply $- 1$ by $- 1$ .

$1 + \cos (- π)$

Step 3.1.4

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

$1 - \cos (0)$

Step 3.1.5

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

$1 - 1 \cdot 1$

Step 3.1.6

Multiply $- 1$ by $1$ .

$1 - 1$

Step 3.2

Subtract $1$ from $1$ .

$0$