Calculus Examples

$\int_{- \frac{π}{4}}^{\frac{π}{4}} \cos (x) d x$

Step 1

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

$\sin (x)]_{- \frac{π}{4}}^{\frac{π}{4}}$

Step 2

Simplify the answer.

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Step 2.1

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

$\sin (\frac{π}{4}) - \sin (- \frac{π}{4})$

Step 2.2

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

$\frac{\sqrt{2}}{2} - \sin (- \frac{π}{4})$

Step 2.3

Simplify.

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Step 2.3.1

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

$\frac{\sqrt{2}}{2} - \sin (\frac{7 π}{4})$

Step 2.3.2

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.

$\frac{\sqrt{2}}{2} - - \sin (\frac{π}{4})$

Step 2.3.3

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

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

Step 2.3.4

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

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Step 2.3.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.4.2

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

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

Step 2.3.5

Combine the numerators over the common denominator.

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

Step 2.3.6

Add $\sqrt{2}$ and $\sqrt{2}$ .

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

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$