Calculus Examples

$\int_{\frac{3 π}{2}}^{2 π} \cos (x) d x$

Step 1

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

$\sin (x)]_{\frac{3 π}{2}}^{2 π}$

Step 2

Evaluate $\sin (x)$ at $2 π$ and at $\frac{3 π}{2}$ .

$\sin (2 π) - \sin (\frac{3 π}{2})$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$\sin (0) - \sin (\frac{3 π}{2})$

Step 3.1.2

The exact value of $\sin (0)$ is $0$ .

$0 - \sin (\frac{3 π}{2})$

Step 3.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$0 - - \sin (\frac{π}{2})$

Step 3.1.4

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

$0 - (- 1 \cdot 1)$

Step 3.1.5

Multiply $- (- 1 \cdot 1)$ .

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Step 3.1.5.1

Multiply $- 1$ by $1$ .

$0 - - 1$

Step 3.1.5.2

Multiply $- 1$ by $- 1$ .

$0 + 1$

Step 3.2

Add $0$ and $1$ .

$1$