Calculus Examples

\int_{0}^{2 π} \sin (x) d x

$\int_{0}^{2 π} \sin (x) d x$

Step 1

The integral of

\sin (x)

$\sin (x)$ with respect to

x

$x$ is

- \cos (x)

$- \cos (x)$ .

- \cos (x)]_{0}^{2 π}

$- \cos (x)]_{0}^{2 π}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate

- \cos (x)

$- \cos (x)$ at

2 π

$2 π$ and at

0

$0$ .

- \cos (2 π) + \cos (0)

$- \cos (2 π) + \cos (0)$

Step 2.2

The exact value of

\cos (0)

$\cos (0)$ is

1

$1$ .

- \cos (2 π) + 1

$- \cos (2 π) + 1$

Step 2.3

Simplify.

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Step 2.3.1

Subtract full rotations of

2 π

$2 π$ until the angle is greater than or equal to

0

$0$ and less than

2 π

$2 π$ .

- \cos (0) + 1

$- \cos (0) + 1$

Step 2.3.2

The exact value of

\cos (0)

$\cos (0)$ is

1

$1$ .

- 1 \cdot 1 + 1

$- 1 \cdot 1 + 1$

Step 2.3.3

Multiply

- 1

$- 1$ by

1

$1$ .

- 1 + 1

$- 1 + 1$

Step 2.3.4

Add

- 1

$- 1$ and

1

$1$ .

0

$0$

0

$0$

0

$0$

\int_{0}^{2 π} \sin x d x

$\int_{0}^{2 π} \sin x d x$

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[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$