Calculus Examples

$\int_{0}^{\frac{3 π}{16}} \frac{\cos (4 x)}{4} + \frac{\sin (4 x)}{4} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{\frac{3 π}{16}} \frac{\cos (4 x)}{4} d x + \int_{0}^{\frac{3 π}{16}} \frac{\sin (4 x)}{4} d x$

Step 2

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{0}^{\frac{3 π}{16}} \cos (4 x) d x + \int_{0}^{\frac{3 π}{16}} \frac{\sin (4 x)}{4} d x$

Step 3

Let $u_{1} = 4 x$ . Then $d u_{1} = 4 d x$ , so $\frac{1}{4} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.1

Let $u_{1} = 4 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 3.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 3.1.3

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

$4 \cdot 1$

Step 3.1.4

Multiply $4$ by $1$ .

$4$

Step 3.2

Substitute the lower limit in for $x$ in $u_{1} = 4 x$ .

$u_{lower} = 4 \cdot 0$

Step 3.3

Multiply $4$ by $0$ .

$u_{lower} = 0$

Step 3.4

Substitute the upper limit in for $x$ in $u_{1} = 4 x$ .

$u_{upper} = 4 \frac{3 π}{16}$

Step 3.5

Cancel the common factor of $4$ .

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Step 3.5.1

Factor $4$ out of $16$ .

$u_{upper} = 4 \frac{3 π}{4 (4)}$

Step 3.5.2

Cancel the common factor.

$u_{upper} = 4 \frac{3 π}{4 \cdot 4}$

Step 3.5.3

Rewrite the expression.

$u_{upper} = \frac{3 π}{4}$

Step 3.6

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

$u_{lower} = 0$

$u_{upper} = \frac{3 π}{4}$

Step 3.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\frac{1}{4} \int_{0}^{\frac{3 π}{4}} \cos (u_{1}) \frac{1}{4} d u_{1} + \int_{0}^{\frac{3 π}{16}} \frac{\sin (4 x)}{4} d x$

Step 4

Combine $\cos (u_{1})$ and $\frac{1}{4}$ .

$\frac{1}{4} \int_{0}^{\frac{3 π}{4}} \frac{\cos (u_{1})}{4} d u_{1} + \int_{0}^{\frac{3 π}{16}} \frac{\sin (4 x)}{4} d x$

Step 5

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} (\frac{1}{4} \int_{0}^{\frac{3 π}{4}} \cos (u_{1}) d u_{1}) + \int_{0}^{\frac{3 π}{16}} \frac{\sin (4 x)}{4} d x$

Step 6

Simplify.

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Step 6.1

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

$\frac{1}{4 \cdot 4} \int_{0}^{\frac{3 π}{4}} \cos (u_{1}) d u_{1} + \int_{0}^{\frac{3 π}{16}} \frac{\sin (4 x)}{4} d x$

Step 6.2

Multiply $4$ by $4$ .

$\frac{1}{16} \int_{0}^{\frac{3 π}{4}} \cos (u_{1}) d u_{1} + \int_{0}^{\frac{3 π}{16}} \frac{\sin (4 x)}{4} d x$

Step 7

The integral of $\cos (u_{1})$ with respect to $u_{1}$ is $\sin (u_{1})$ .

$\frac{1}{16} \sin (u_{1})]_{0}^{\frac{3 π}{4}} + \int_{0}^{\frac{3 π}{16}} \frac{\sin (4 x)}{4} d x$

Step 8

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{16} \sin (u_{1})]_{0}^{\frac{3 π}{4}} + \frac{1}{4} \int_{0}^{\frac{3 π}{16}} \sin (4 x) d x$

Step 9

Let $u_{2} = 4 x$ . Then $d u_{2} = 4 d x$ , so $\frac{1}{4} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 9.1

Let $u_{2} = 4 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 9.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 9.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 9.1.3

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

$4 \cdot 1$

Step 9.1.4

Multiply $4$ by $1$ .

$4$

Step 9.2

Substitute the lower limit in for $x$ in $u_{2} = 4 x$ .

$u_{lower} = 4 \cdot 0$

Step 9.3

Multiply $4$ by $0$ .

$u_{lower} = 0$

Step 9.4

Substitute the upper limit in for $x$ in $u_{2} = 4 x$ .

$u_{upper} = 4 \frac{3 π}{16}$

Step 9.5

Cancel the common factor of $4$ .

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Step 9.5.1

Factor $4$ out of $16$ .

$u_{upper} = 4 \frac{3 π}{4 (4)}$

Step 9.5.2

Cancel the common factor.

$u_{upper} = 4 \frac{3 π}{4 \cdot 4}$

Step 9.5.3

Rewrite the expression.

$u_{upper} = \frac{3 π}{4}$

Step 9.6

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

$u_{lower} = 0$

$u_{upper} = \frac{3 π}{4}$

Step 9.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{1}{16} \sin (u_{1})]_{0}^{\frac{3 π}{4}} + \frac{1}{4} \int_{0}^{\frac{3 π}{4}} \sin (u_{2}) \frac{1}{4} d u_{2}$

Step 10

Combine $\sin (u_{2})$ and $\frac{1}{4}$ .

$\frac{1}{16} \sin (u_{1})]_{0}^{\frac{3 π}{4}} + \frac{1}{4} \int_{0}^{\frac{3 π}{4}} \frac{\sin (u_{2})}{4} d u_{2}$

Step 11

Since $\frac{1}{4}$ is constant with respect to $u_{2}$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{16} \sin (u_{1})]_{0}^{\frac{3 π}{4}} + \frac{1}{4} (\frac{1}{4} \int_{0}^{\frac{3 π}{4}} \sin (u_{2}) d u_{2})$

Step 12

Simplify.

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Step 12.1

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

$\frac{1}{16} \sin (u_{1})]_{0}^{\frac{3 π}{4}} + \frac{1}{4 \cdot 4} \int_{0}^{\frac{3 π}{4}} \sin (u_{2}) d u_{2}$

Step 12.2

Multiply $4$ by $4$ .

$\frac{1}{16} \sin (u_{1})]_{0}^{\frac{3 π}{4}} + \frac{1}{16} \int_{0}^{\frac{3 π}{4}} \sin (u_{2}) d u_{2}$

Step 13

The integral of $\sin (u_{2})$ with respect to $u_{2}$ is $- \cos (u_{2})$ .

$\frac{1}{16} \sin (u_{1})]_{0}^{\frac{3 π}{4}} + \frac{1}{16} - \cos (u_{2})]_{0}^{\frac{3 π}{4}}$

Step 14

Substitute and simplify.

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Step 14.1

Evaluate $\sin (u_{1})$ at $\frac{3 π}{4}$ and at $0$ .

$\frac{1}{16} ((\sin (\frac{3 π}{4})) - \sin (0)) + \frac{1}{16} - \cos (u_{2})]_{0}^{\frac{3 π}{4}}$

Step 14.2

Evaluate $- \cos (u_{2})$ at $\frac{3 π}{4}$ and at $0$ .

$\frac{1}{16} ((\sin (\frac{3 π}{4})) - \sin (0)) + \frac{1}{16} ((- \cos (\frac{3 π}{4})) + \cos (0))$

Step 14.3

Remove parentheses.

$\frac{1}{16} (\sin (\frac{3 π}{4}) - \sin (0)) + \frac{1}{16} (- \cos (\frac{3 π}{4}) + \cos (0))$

Step 15

Simplify.

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Step 15.1

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

$\frac{1}{16} (\sin (\frac{3 π}{4}) - 0) + \frac{1}{16} (- \cos (\frac{3 π}{4}) + \cos (0))$

Step 15.2

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

$\frac{1}{16} (\sin (\frac{3 π}{4}) - 0) + \frac{1}{16} (- \cos (\frac{3 π}{4}) + 1)$

Step 15.3

Multiply $- 1$ by $0$ .

$\frac{1}{16} (\sin (\frac{3 π}{4}) + 0) + \frac{1}{16} (- \cos (\frac{3 π}{4}) + 1)$

Step 15.4

Add $\sin (\frac{3 π}{4})$ and $0$ .

$\frac{1}{16} \sin (\frac{3 π}{4}) + \frac{1}{16} (- \cos (\frac{3 π}{4}) + 1)$

Step 15.5

Combine $\frac{1}{16}$ and $\sin (\frac{3 π}{4})$ .

$\frac{\sin (\frac{3 π}{4})}{16} + \frac{1}{16} (- \cos (\frac{3 π}{4}) + 1)$

Step 15.6

To write $\frac{1}{16} (- \cos (\frac{3 π}{4}) + 1)$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{\sin (\frac{3 π}{4})}{16} + \frac{1}{16} (- \cos (\frac{3 π}{4}) + 1) \cdot \frac{16}{16}$

Step 15.7

Combine $\frac{1}{16} (- \cos (\frac{3 π}{4}) + 1)$ and $\frac{16}{16}$ .

$\frac{\sin (\frac{3 π}{4})}{16} + \frac{\frac{1}{16} (- \cos (\frac{3 π}{4}) + 1) \cdot 16}{16}$

Step 15.8

Combine the numerators over the common denominator.

$\frac{\sin (\frac{3 π}{4}) + \frac{1}{16} (- \cos (\frac{3 π}{4}) + 1) \cdot 16}{16}$

Step 15.9

Combine $16$ and $\frac{1}{16}$ .

$\frac{\sin (\frac{3 π}{4}) + \frac{16}{16} (- \cos (\frac{3 π}{4}) + 1)}{16}$

Step 15.10

Cancel the common factor of $16$ .

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Step 15.10.1

Cancel the common factor.

$\frac{\sin (\frac{3 π}{4}) + \frac{16}{16} (- \cos (\frac{3 π}{4}) + 1)}{16}$

Step 15.10.2

Rewrite the expression.

$\frac{\sin (\frac{3 π}{4}) + 1 (- \cos (\frac{3 π}{4}) + 1)}{16}$

Step 15.11

Multiply $- \cos (\frac{3 π}{4}) + 1$ by $1$ .

$\frac{\sin (\frac{3 π}{4}) - \cos (\frac{3 π}{4}) + 1}{16}$

Step 16

Simplify.

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Step 16.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{\sin (\frac{3 π}{4}) - - \cos (\frac{π}{4}) + 1}{16}$

Step 16.2

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

$\frac{\sin (\frac{3 π}{4}) - - \frac{\sqrt{2}}{2} + 1}{16}$

Step 16.3

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 16.3.1

Multiply $- 1$ by $- 1$ .

$\frac{\sin (\frac{3 π}{4}) + 1 \frac{\sqrt{2}}{2} + 1}{16}$

Step 16.3.2

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

$\frac{\sin (\frac{3 π}{4}) + \frac{\sqrt{2}}{2} + 1}{16}$

Step 16.4

Simplify the numerator.

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Step 16.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{\sin (\frac{π}{4}) + \frac{\sqrt{2}}{2} + 1}{16}$

Step 16.4.2

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

$\frac{\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + 1}{16}$

Step 16.4.3

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{2} + \sqrt{2}}{2} + 1}{16}$

Step 16.4.4

Add $\sqrt{2}$ and $\sqrt{2}$ .

$\frac{\frac{2 \sqrt{2}}{2} + 1}{16}$

Step 16.4.5

Cancel the common factor of $2$ .

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Step 16.4.5.1

Cancel the common factor.

$\frac{\frac{2 \sqrt{2}}{2} + 1}{16}$

Step 16.4.5.2

Divide $\sqrt{2}$ by $1$ .

$\frac{\sqrt{2} + 1}{16}$

Step 17

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2} + 1}{16}$

Decimal Form:

$0.15088834 \dots$