Calculus Examples

$\int_{\frac{π}{2}}^{\frac{3 π}{4}} \sin (x) d x$

Step 1

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

$- \cos (x)]_{\frac{π}{2}}^{\frac{3 π}{4}}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $- \cos (x)$ at $\frac{3 π}{4}$ and at $\frac{π}{2}$ .

$- \cos (\frac{3 π}{4}) + \cos (\frac{π}{2})$

Step 2.2

Simplify.

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Step 2.2.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- \cos (\frac{3 π}{4}) + 0$

Step 2.2.2

Add $- \cos (\frac{3 π}{4})$ and $0$ .

$- \cos (\frac{3 π}{4})$

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.

$- - \cos (\frac{π}{4})$

Step 2.3.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- - \frac{\sqrt{2}}{2}$

Step 2.3.3

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 2.3.3.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{2}}{2}$

Step 2.3.3.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$\frac{\sqrt{2}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{2}$

Decimal Form:

$0.70710678 \dots$