Calculus Examples

$\int_{π}^{2 π} \sin (x) d x$

Step 1

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

$- \cos (x)]_{π}^{2 π}$

Step 2

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

$- \cos (2 π) + \cos (π)$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$- \cos (0) + \cos (π)$

Step 3.1.2

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

$- 1 \cdot 1 + \cos (π)$

Step 3.1.3

Multiply $- 1$ 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 $- 1$ .

$- 2$