Calculus Examples

$\int_{0}^{\frac{π}{8}} \sin (2 x) d x$

Step 1

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

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Step 1.1

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

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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 $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 1.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 1.4

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

$u_{upper} = 2 \frac{π}{8}$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $8$ .

$u_{upper} = 2 \frac{π}{2 (4)}$

Step 1.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 4}$

Step 1.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{4}$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{4}$

Step 1.7

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

$\int_{0}^{\frac{π}{4}} \sin (u) \frac{1}{2} d u$

Step 2

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

$\int_{0}^{\frac{π}{4}} \frac{\sin (u)}{2} d u$

Step 3

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

$\frac{1}{2} \int_{0}^{\frac{π}{4}} \sin (u) d u$

Step 4

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

$\frac{1}{2} - \cos (u)]_{0}^{\frac{π}{4}}$

Step 5

Evaluate $- \cos (u)$ at $\frac{π}{4}$ and at $0$ .

$\frac{1}{2} (- \cos (\frac{π}{4}) + \cos (0))$

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{2} (- \frac{\sqrt{2}}{2} + \cos (0))$

Step 6.2

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

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

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $\frac{1}{2} (- \frac{\sqrt{2}}{2})$ .

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Step 7.2.1

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

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

Step 7.2.2

Multiply $2$ by $2$ .

$- \frac{\sqrt{2}}{4} + \frac{1}{2} \cdot 1$

Step 7.3

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

$- \frac{\sqrt{2}}{4} + \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{2}}{4} + \frac{1}{2}$

Decimal Form:

$0.14644660 \dots$