Calculus Examples

$\int_{\frac{π}{3}}^{π} \sin (x) d x$

Step 1

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

$- \cos (x)]_{\frac{π}{3}}^{π}$

Step 2

Simplify the answer.

Tap for more steps...

Step 2.1

Evaluate $- \cos (x)$ at $π$ and at $\frac{π}{3}$ .

$- \cos (π) + \cos (\frac{π}{3})$

Step 2.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$- \cos (π) + \frac{1}{2}$

Step 2.3

Simplify.

Tap for more steps...

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.

$- - \cos (0) + \frac{1}{2}$

Step 2.3.2

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

$- (- 1 \cdot 1) + \frac{1}{2}$

Step 2.3.3

Multiply $- 1$ by $1$ .

$- - 1 + \frac{1}{2}$

Step 2.3.4

Multiply $- 1$ by $- 1$ .

$1 + \frac{1}{2}$

Step 2.3.5

Write $1$ as a fraction with a common denominator.

$\frac{2}{2} + \frac{1}{2}$

Step 2.3.6

Combine the numerators over the common denominator.

$\frac{2 + 1}{2}$

Step 2.3.7

Add $2$ and $1$ .

$\frac{3}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$

Mixed Number Form:

$1 \frac{1}{2}$