Calculus Examples

$\int_{0}^{1} \cos (\frac{π t}{2}) d t$

Step 1

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

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Differentiate $\frac{π t}{2}$ .

$\frac{d}{d t} [\frac{π t}{2}]$

Step 1.1.2

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

$\frac{π}{2} \frac{d}{d t} [t]$

Step 1.1.3

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

$\frac{π}{2} \cdot 1$

Step 1.1.4

Multiply $\frac{π}{2}$ by $1$ .

$\frac{π}{2}$

Step 1.2

Substitute the lower limit in for $t$ in $u = \frac{π t}{2}$ .

$u_{lower} = \frac{π \cdot 0}{2}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Cancel the common factor of $0$ and $2$ .

Tap for more steps...

Step 1.3.1.1

Factor $2$ out of $π \cdot 0$ .

$u_{lower} = \frac{2 (π \cdot 0)}{2}$

Step 1.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.3.1.2.1

Factor $2$ out of $2$ .

$u_{lower} = \frac{2 (π \cdot 0)}{2 (1)}$

Step 1.3.1.2.2

Cancel the common factor.

$u_{lower} = \frac{2 (π \cdot 0)}{2 \cdot 1}$

Step 1.3.1.2.3

Rewrite the expression.

$u_{lower} = \frac{π \cdot 0}{1}$

Step 1.3.1.2.4

Divide $π \cdot 0$ by $1$ .

$u_{lower} = π \cdot 0$

Step 1.3.2

Multiply $π$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $t$ in $u = \frac{π t}{2}$ .

$u_{upper} = \frac{π \cdot 1}{2}$

Step 1.5

Multiply $π$ by $1$ .

$u_{upper} = \frac{π}{2}$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{2}$

Step 1.7

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

$\int_{0}^{\frac{π}{2}} \cos (u) \frac{1}{\frac{π}{2}} d u$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{π}{2}$ .

$\int_{0}^{\frac{π}{2}} \cos (u) (1 \frac{2}{π}) d u$

Step 2.2

Multiply $\frac{2}{π}$ by $1$ .

$\int_{0}^{\frac{π}{2}} \cos (u) \frac{2}{π} d u$

Step 2.3

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

$\int_{0}^{\frac{π}{2}} \frac{\cos (u) \cdot 2}{π} d u$

Step 2.4

Move $2$ to the left of $\cos (u)$ .

$\int_{0}^{\frac{π}{2}} \frac{2 \cos (u)}{π} d u$

Step 3

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

$\frac{2}{π} \int_{0}^{\frac{π}{2}} \cos (u) d u$

Step 4

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

$\frac{2}{π} \sin (u)]_{0}^{\frac{π}{2}}$

Step 5

Evaluate $\sin (u)$ at $\frac{π}{2}$ and at $0$ .

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

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

$\frac{2}{π} (1 - \sin (0))$

Step 6.2

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

$\frac{2}{π} (1 - 0)$

Step 6.3

Multiply $- 1$ by $0$ .

$\frac{2}{π} (1 + 0)$

Step 6.4

Add $1$ and $0$ .

$\frac{2}{π} \cdot 1$

Step 6.5

Multiply $\frac{2}{π}$ by $1$ .

$\frac{2}{π}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{π}$

Decimal Form:

$0.63661977 \dots$