Calculus Examples

$\int_{\frac{π}{4}}^{π} \sin (x) d x$

Step 1

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

$- \cos (x)]_{\frac{π}{4}}^{π}$

Step 2

Simplify the answer.

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Step 2.1

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

$- \cos (π) + \cos (\frac{π}{4})$

Step 2.2

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

$- \cos (π) + \frac{\sqrt{2}}{2}$

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.

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $- 1$ by $1$ .

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

Step 2.3.4

Multiply $- 1$ by $- 1$ .

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.70710678 \dots$