Calculus Examples

\int_{\frac{π}{2}}^{π} 5 cos (x) d x

$\int_{\frac{π}{2}}^{π} 5 \cos (x) d x$

Step 1

Since

5

$5$ is constant with respect to

x

$x$ , move

5

$5$ out of the integral.

5 \int_{\frac{π}{2}}^{π} cos (x) d x

$5 \int_{\frac{π}{2}}^{π} \cos (x) d x$

Step 2

The integral of

cos (x)

$\cos (x)$ with respect to

x

$x$ is

sin (x)

$\sin (x)$ .

5 (sin (x)]_{\frac{π}{2}}^{π})

$5 (\sin (x)]_{\frac{π}{2}}^{π})$

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Evaluate

sin (x)

$\sin (x)$ at

π

$π$ and at

\frac{π}{2}

$\frac{π}{2}$ .

5 (sin (π) - sin (\frac{π}{2}))

$5 (\sin (π) - \sin (\frac{π}{2}))$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

The exact value of

sin (\frac{π}{2})

$\sin (\frac{π}{2})$ is

1

$1$ .

5 (sin (π) - 1 \cdot 1)

$5 (\sin (π) - 1 \cdot 1)$

Step 3.2.2

Multiply

- 1

$- 1$ by

1

$1$ .

5 (sin (π) - 1)

$5 (\sin (π) - 1)$

5 (sin (π) - 1)

$5 (\sin (π) - 1)$

Step 3.3

Simplify.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

5 (sin (0) - 1)

$5 (\sin (0) - 1)$

Step 3.3.1.2

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

5 (0 - 1)

$5 (0 - 1)$

5 (0 - 1)

$5 (0 - 1)$

Step 3.3.2

Subtract

1

$1$ from

0

$0$ .

5 \cdot - 1

$5 \cdot - 1$

Step 3.3.3

Multiply

5

$5$ by

- 1

$- 1$ .

- 5

$- 5$

- 5

$- 5$

- 5

$- 5$