Calculus Examples

$\int_{0}^{2 π} \sin (x) d x$

Step 1

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

$- \cos (x)]_{0}^{2 π}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $- \cos (x)$ at $2 π$ and at $0$ .

$- \cos (2 π) + \cos (0)$

Step 2.2

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

$- \cos (2 π) + 1$

Step 2.3

Simplify.

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Step 2.3.1

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

$- \cos (0) + 1$

Step 2.3.2

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

$- 1 \cdot 1 + 1$

Step 2.3.3

Multiply $- 1$ by $1$ .

$- 1 + 1$

Step 2.3.4

Add $- 1$ and $1$ .

$0$