Calculus Examples

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

Step 1

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \frac{π}{4}$

Step 1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.3.1

Factor $2$ out of $4$ .

$u_{lower} = 2 \frac{π}{2 (2)}$

Step 1.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{π}{2 \cdot 2}$

Step 1.3.3

Rewrite the expression.

$u_{lower} = \frac{π}{2}$

Step 1.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \frac{π}{2}$

Step 1.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 1.5.2

Rewrite the expression.

$u_{upper} = π$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = \frac{π}{2}$

$u_{upper} = π$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{\frac{π}{2}}^{π} \cos (u) \frac{1}{2} d u$

Step 2

Combine $\cos (u)$ and $\frac{1}{2}$ .

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

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{\frac{π}{2}}^{π} \cos (u) d u$

Step 4

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

$\frac{1}{2} \sin (u)]_{\frac{π}{2}}^{π}$

Step 5

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

$\frac{1}{2} (\sin (π) - \sin (\frac{π}{2}))$

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

$\frac{1}{2} (\sin (π) - 1 \cdot 1)$

Step 6.2

Multiply $- 1$ by $1$ .

$\frac{1}{2} (\sin (π) - 1)$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{1}{2} (\sin (0) - 1)$

Step 7.1.2

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

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

Step 7.2

Subtract $1$ from $0$ .

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

Step 7.3

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{- 1}{2}$

Step 7.4

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$