Calculus Examples

$\int_{0}^{\frac{π}{4}} \cos^{2} (x) d x$

Step 1

Use the half-angle formula to rewrite $\cos^{2} (x)$ as $\frac{1 + \cos (2 x)}{2}$ .

$\int_{0}^{\frac{π}{4}} \frac{1 + \cos (2 x)}{2} d x$

Step 2

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

$\frac{1}{2} \int_{0}^{\frac{π}{4}} 1 + \cos (2 x) d x$

Step 3

Split the single integral into multiple integrals.

$\frac{1}{2} (\int_{0}^{\frac{π}{4}} d x + \int_{0}^{\frac{π}{4}} \cos (2 x) d x)$

Step 4

Apply the constant rule.

$\frac{1}{2} (x]_{0}^{\frac{π}{4}} + \int_{0}^{\frac{π}{4}} \cos (2 x) d x)$

Step 5

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 5.1

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

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Step 5.1.1

Differentiate $2 x$ .

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

Step 5.1.2

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 5.1.3

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 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

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

$u_{lower} = 2 \cdot 0$

Step 5.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 5.4

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

$u_{upper} = 2 \frac{π}{4}$

Step 5.5

Cancel the common factor of $2$ .

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Step 5.5.1

Factor $2$ out of $4$ .

$u_{upper} = 2 \frac{π}{2 (2)}$

Step 5.5.2

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2 \cdot 2}$

Step 5.5.3

Rewrite the expression.

$u_{upper} = \frac{π}{2}$

Step 5.6

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 5.7

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

$\frac{1}{2} (x]_{0}^{\frac{π}{4}} + \int_{0}^{\frac{π}{2}} \cos (u) \frac{1}{2} d u)$

Step 6

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

$\frac{1}{2} (x]_{0}^{\frac{π}{4}} + \int_{0}^{\frac{π}{2}} \frac{\cos (u)}{2} d u)$

Step 7

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

$\frac{1}{2} (x]_{0}^{\frac{π}{4}} + \frac{1}{2} \int_{0}^{\frac{π}{2}} \cos (u) d u)$

Step 8

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

$\frac{1}{2} (x]_{0}^{\frac{π}{4}} + \frac{1}{2} \sin (u)]_{0}^{\frac{π}{2}})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $x$ at $\frac{π}{4}$ and at $0$ .

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

Step 9.2

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

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

Step 9.3

Add $\frac{π}{4}$ and $0$ .

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

Multiply $- 1$ by $0$ .

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

Step 10.4

Add $1$ and $0$ .

$\frac{1}{2} (\frac{π}{4} + \frac{1}{2} \cdot 1)$

Step 10.5

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

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

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\frac{1}{2} \cdot \frac{π}{4} + \frac{1}{2} \cdot \frac{1}{2}$

Step 11.2

Multiply $\frac{1}{2} \cdot \frac{π}{4}$ .

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Step 11.2.1

Multiply $\frac{1}{2}$ by $\frac{π}{4}$ .

$\frac{π}{2 \cdot 4} + \frac{1}{2} \cdot \frac{1}{2}$

Step 11.2.2

Multiply $2$ by $4$ .

$\frac{π}{8} + \frac{1}{2} \cdot \frac{1}{2}$

Step 11.3

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

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Step 11.3.1

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

$\frac{π}{8} + \frac{1}{2 \cdot 2}$

Step 11.3.2

Multiply $2$ by $2$ .

$\frac{π}{8} + \frac{1}{4}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{8} + \frac{1}{4}$

Decimal Form:

$0.64269908 \dots$