Calculus Examples

\int_{0}^{π} \sin (x) d x

$\int_{0}^{π} \sin (x) d x$

Step 1

The integral of

\sin (x)

$\sin (x)$ with respect to

x

$x$ is

- \cos (x)

$- \cos (x)$ .

- \cos (x)]_{0}^{π}

$- \cos (x)]_{0}^{π}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate

- \cos (x)

$- \cos (x)$ at

π

$π$ and at

0

$0$ .

- \cos (π) + \cos (0)

$- \cos (π) + \cos (0)$

Step 2.2

The exact value of

\cos (0)

$\cos (0)$ is

1

$1$ .

- \cos (π) + 1

$- \cos (π) + 1$

Step 2.3

Simplify.

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Step 2.3.1

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) + 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

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

1 + 1

$1 + 1$

Step 2.3.5

Add

1

$1$ and

1

$1$ .

2

$2$

2

$2$

2

$2$

\int_{0}^{π} \sin x d x

$\int_{0}^{π} \sin x d x$

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