Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $π x$ .

$\frac{d}{d x} [π x]$

Step 1.1.2

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

$π \frac{d}{d x} [x]$

Step 1.1.3

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

$π \cdot 1$

Step 1.1.4

Multiply $π$ by $1$ .

$π$

Step 1.2

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

$u_{lower} = π \cdot 0$

Step 1.3

Multiply $π$ by $0$ .

$u_{lower} = 0$

Step 1.4

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

$u_{upper} = π \frac{1}{2}$

Step 1.5

Combine $π$ and $\frac{1}{2}$ .

$u_{upper} = \frac{π}{2}$

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.

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

Step 2

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

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

Step 3

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

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

Step 4

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

$\frac{1}{π} \sin (u)]_{0}^{\frac{π}{2}}$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{π} (1 - \sin (0))$

Step 6.2

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

$\frac{1}{π} (1 - 0)$

Step 6.3

Multiply $- 1$ by $0$ .

$\frac{1}{π} (1 + 0)$

Step 6.4

Add $1$ and $0$ .

$\frac{1}{π} \cdot 1$

Step 6.5

Multiply $\frac{1}{π}$ by $1$ .

$\frac{1}{π}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{π}$

Decimal Form:

$0.31830988 \dots$