Calculus Examples

$\int_{0}^{\frac{π}{2}} \cos (\frac{2 x}{3}) d x$

Step 1

Let $u = \frac{2 x}{3}$ . Then $d u = \frac{2}{3} d x$ , so $\frac{3}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{2 x}{3}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\frac{2 x}{3}$ .

$\frac{d}{d x} [\frac{2 x}{3}]$

Step 1.1.2

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 x}{3}$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [x]$ .

$\frac{2}{3} \frac{d}{d x} [x]$

Step 1.1.3

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

$\frac{2}{3} \cdot 1$

Step 1.1.4

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{2 x}{3}$ .

$u_{lower} = \frac{2 \cdot 0}{3}$

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of $0$ and $3$ .

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Step 1.3.1.1

Factor $3$ out of $2 \cdot 0$ .

$u_{lower} = \frac{3 (2 \cdot 0)}{3}$

Step 1.3.1.2

Cancel the common factors.

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Step 1.3.1.2.1

Factor $3$ out of $3$ .

$u_{lower} = \frac{3 (2 \cdot 0)}{3 (1)}$

Step 1.3.1.2.2

Cancel the common factor.

$u_{lower} = \frac{3 (2 \cdot 0)}{3 \cdot 1}$

Step 1.3.1.2.3

Rewrite the expression.

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

Step 1.3.1.2.4

Divide $2 \cdot 0$ by $1$ .

$u_{lower} = 2 \cdot 0$

Step 1.3.2

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{2 x}{3}$ .

$u_{upper} = \frac{2 \frac{π}{2}}{3}$

Step 1.5

Simplify.

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Step 1.5.1

Combine $2$ and $\frac{π}{2}$ .

$u_{upper} = \frac{\frac{2 π}{2}}{3}$

Step 1.5.2

Reduce the expression by cancelling the common factors.

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Step 1.5.2.1

Reduce the expression $\frac{2 π}{2}$ by cancelling the common factors.

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Step 1.5.2.1.1

Cancel the common factor.

$u_{upper} = \frac{\frac{2 π}{2}}{3}$

Step 1.5.2.1.2

Rewrite the expression.

$u_{upper} = \frac{\frac{π}{1}}{3}$

Step 1.5.2.2

Divide $π$ by $1$ .

$u_{upper} = \frac{π}{3}$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{3}$

Step 1.7

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

$\int_{0}^{\frac{π}{3}} \cos (u) \frac{1}{\frac{2}{3}} d u$

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{2}{3}$ .

$\int_{0}^{\frac{π}{3}} \cos (u) (1 (\frac{3}{2})) d u$

Step 2.2

Multiply $\frac{3}{2}$ by $1$ .

$\int_{0}^{\frac{π}{3}} \cos (u) \frac{3}{2} d u$

Step 2.3

Combine $\cos (u)$ and $\frac{3}{2}$ .

$\int_{0}^{\frac{π}{3}} \frac{\cos (u) \cdot 3}{2} d u$

Step 2.4

Move $3$ to the left of $\cos (u)$ .

$\int_{0}^{\frac{π}{3}} \frac{3 \cos (u)}{2} d u$

Step 3

Since $\frac{3}{2}$ is constant with respect to $u$ , move $\frac{3}{2}$ out of the integral.

$\frac{3}{2} \int_{0}^{\frac{π}{3}} \cos (u) d u$

Step 4

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

$\frac{3}{2} \sin (u)]_{0}^{\frac{π}{3}}$

Step 5

Evaluate $\sin (u)$ at $\frac{π}{3}$ and at $0$ .

$\frac{3}{2} (\sin (\frac{π}{3}) - \sin (0))$

Step 6

Simplify.

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Step 6.1

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{3}{2} (\frac{\sqrt{3}}{2} - \sin (0))$

Step 6.2

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

$\frac{3}{2} (\frac{\sqrt{3}}{2} - 0)$

Step 6.3

Multiply $- 1$ by $0$ .

$\frac{3}{2} (\frac{\sqrt{3}}{2} + 0)$

Step 6.4

Add $\frac{\sqrt{3}}{2}$ and $0$ .

$\frac{3}{2} \cdot \frac{\sqrt{3}}{2}$

Step 6.5

Multiply $\frac{3}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{3 \sqrt{3}}{2 \cdot 2}$

Step 6.6

Multiply $2$ by $2$ .

$\frac{3 \sqrt{3}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt{3}}{4}$

Decimal Form:

$1.29903810 \dots$