Calculus Examples

$\int \sin (2 x) + \cos^{2} (2 x) d x$

Step 1

Split the single integral into multiple integrals.

$\int \sin (2 x) d x + \int \cos^{2} (2 x) d x$

Step 2

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

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Step 2.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $2 x$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \sin (u_{1}) \frac{1}{2} d u_{1} + \int \cos^{2} (2 x) d x$

Step 3

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

$\int \frac{\sin (u_{1})}{2} d u_{1} + \int \cos^{2} (2 x) d x$

Step 4

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

$\frac{1}{2} \int \sin (u_{1}) d u_{1} + \int \cos^{2} (2 x) d x$

Step 5

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

$\frac{1}{2} (- \cos (u_{1}) + C) + \int \cos^{2} (2 x) d x$

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $2 x$ .

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

Step 6.1.2

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 6.1.3

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 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} (- \cos (u_{1}) + C) + \int \cos^{2} (u_{2}) \frac{1}{2} d u_{2}$

Step 7

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

$\frac{1}{2} (- \cos (u_{1}) + C) + \int \frac{\cos^{2} (u_{2})}{2} d u_{2}$

Step 8

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

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{2} \int \cos^{2} (u_{2}) d u_{2}$

Step 9

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

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{2} \int \frac{1 + \cos (2 u_{2})}{2} d u_{2}$

Step 10

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

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{2} (\frac{1}{2} \int 1 + \cos (2 u_{2}) d u_{2})$

Step 11

Simplify.

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Step 11.1

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

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{2 \cdot 2} \int 1 + \cos (2 u_{2}) d u_{2}$

Step 11.2

Multiply $2$ by $2$ .

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{4} \int 1 + \cos (2 u_{2}) d u_{2}$

Step 12

Split the single integral into multiple integrals.

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{4} (\int d u_{2} + \int \cos (2 u_{2}) d u_{2})$

Step 13

Apply the constant rule.

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{4} (u_{2} + C + \int \cos (2 u_{2}) d u_{2})$

Step 14

Let $u_{3} = 2 u_{2}$ . Then $d u_{3} = 2 d u_{2}$ , so $\frac{1}{2} d u_{3} = d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 14.1

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

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Step 14.1.1

Differentiate $2 u_{2}$ .

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

Step 14.1.2

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

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

Step 14.1.3

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 14.1.4

Multiply $2$ by $1$ .

$2$

Step 14.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{4} (u_{2} + C + \int \cos (u_{3}) \frac{1}{2} d u_{3})$

Step 15

Combine $\cos (u_{3})$ and $\frac{1}{2}$ .

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{4} (u_{2} + C + \int \frac{\cos (u_{3})}{2} d u_{3})$

Step 16

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{4} (u_{2} + C + \frac{1}{2} \int \cos (u_{3}) d u_{3})$

Step 17

The integral of $\cos (u_{3})$ with respect to $u_{3}$ is $\sin (u_{3})$ .

$\frac{1}{2} (- \cos (u_{1}) + C) + \frac{1}{4} (u_{2} + C + \frac{1}{2} (\sin (u_{3}) + C))$

Step 18

Simplify.

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Step 18.1

Simplify.

$- \frac{\cos (u_{1})}{2} + \frac{1}{4} (u_{2} + \frac{1}{2} \sin (u_{3})) + C$

Step 18.2

Simplify.

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Step 18.2.1

To write $\frac{1}{4} (u_{2} + \frac{1}{2} \sin (u_{3}))$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{\cos (u_{1})}{2} + \frac{1}{4} (u_{2} + \frac{1}{2} \sin (u_{3})) \cdot \frac{2}{2} + C$

Step 18.2.2

Combine $\frac{1}{4} (u_{2} + \frac{1}{2} \sin (u_{3}))$ and $\frac{2}{2}$ .

$- \frac{\cos (u_{1})}{2} + \frac{\frac{1}{4} (u_{2} + \frac{1}{2} \sin (u_{3})) \cdot 2}{2} + C$

Step 18.2.3

Combine the numerators over the common denominator.

$\frac{- \cos (u_{1}) + \frac{1}{4} (u_{2} + \frac{1}{2} \sin (u_{3})) \cdot 2}{2} + C$

Step 18.2.4

Combine $\frac{1}{2}$ and $\sin (u_{3})$ .

$\frac{- \cos (u_{1}) + \frac{1}{4} (u_{2} + \frac{\sin (u_{3})}{2}) \cdot 2}{2} + C$

Step 18.2.5

Combine $2$ and $\frac{1}{4}$ .

$\frac{- \cos (u_{1}) + \frac{2}{4} (u_{2} + \frac{\sin (u_{3})}{2})}{2} + C$

Step 18.2.6

Cancel the common factor of $2$ and $4$ .

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Step 18.2.6.1

Factor $2$ out of $2$ .

$\frac{- \cos (u_{1}) + \frac{2 (1)}{4} (u_{2} + \frac{\sin (u_{3})}{2})}{2} + C$

Step 18.2.6.2

Cancel the common factors.

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Step 18.2.6.2.1

Factor $2$ out of $4$ .

$\frac{- \cos (u_{1}) + \frac{2 \cdot 1}{2 \cdot 2} (u_{2} + \frac{\sin (u_{3})}{2})}{2} + C$

Step 18.2.6.2.2

Cancel the common factor.

$\frac{- \cos (u_{1}) + \frac{2 \cdot 1}{2 \cdot 2} (u_{2} + \frac{\sin (u_{3})}{2})}{2} + C$

Step 18.2.6.2.3

Rewrite the expression.

$\frac{- \cos (u_{1}) + \frac{1}{2} (u_{2} + \frac{\sin (u_{3})}{2})}{2} + C$

Step 19

Substitute back in for each integration substitution variable.

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Step 19.1

Replace all occurrences of $u_{1}$ with $2 x$ .

$\frac{- \cos (2 x) + \frac{1}{2} (u_{2} + \frac{\sin (u_{3})}{2})}{2} + C$

Step 19.2

Replace all occurrences of $u_{2}$ with $2 x$ .

$\frac{- \cos (2 x) + \frac{1}{2} (2 x + \frac{\sin (u_{3})}{2})}{2} + C$

Step 19.3

Replace all occurrences of $u_{3}$ with $2 u_{2}$ .

$\frac{- \cos (2 x) + \frac{1}{2} (2 x + \frac{\sin (2 u_{2})}{2})}{2} + C$

Step 19.4

Replace all occurrences of $u_{2}$ with $2 x$ .

$\frac{- \cos (2 x) + \frac{1}{2} (2 x + \frac{\sin (2 (2 x))}{2})}{2} + C$

Step 20

Simplify.

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Step 20.1

Multiply $2$ by $2$ .

$\frac{- \cos (2 x) + \frac{1}{2} (2 x + \frac{\sin (4 x)}{2})}{2} + C$

Step 20.2

Apply the distributive property.

$\frac{- \cos (2 x) + \frac{1}{2} (2 x) + \frac{1}{2} \cdot \frac{\sin (4 x)}{2}}{2} + C$

Step 20.3

Cancel the common factor of $2$ .

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Step 20.3.1

Factor $2$ out of $2 x$ .

$\frac{- \cos (2 x) + \frac{1}{2} (2 (x)) + \frac{1}{2} \cdot \frac{\sin (4 x)}{2}}{2} + C$

Step 20.3.2

Cancel the common factor.

$\frac{- \cos (2 x) + \frac{1}{2} (2 x) + \frac{1}{2} \cdot \frac{\sin (4 x)}{2}}{2} + C$

Step 20.3.3

Rewrite the expression.

$\frac{- \cos (2 x) + x + \frac{1}{2} \cdot \frac{\sin (4 x)}{2}}{2} + C$

Step 20.4

Multiply $\frac{1}{2} \cdot \frac{\sin (4 x)}{2}$ .

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Step 20.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (4 x)}{2}$ .

$\frac{- \cos (2 x) + x + \frac{\sin (4 x)}{2 \cdot 2}}{2} + C$

Step 20.4.2

Multiply $2$ by $2$ .

$\frac{- \cos (2 x) + x + \frac{\sin (4 x)}{4}}{2} + C$

Step 21

Simplify.

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Step 21.1

Factor $- 1$ out of $- \cos (2 x)$ .

$\frac{- (\cos (2 x)) + x + \frac{\sin (4 x)}{4}}{2} + C$

Step 21.2

Factor $- 1$ out of $x$ .

$\frac{- (\cos (2 x)) - 1 (- x) + \frac{\sin (4 x)}{4}}{2} + C$

Step 21.3

Factor $- 1$ out of $- (\cos (2 x)) - 1 (- x)$ .

$\frac{- (\cos (2 x) - x) + \frac{\sin (4 x)}{4}}{2} + C$

Step 21.4

Factor $- 1$ out of $\frac{\sin (4 x)}{4}$ .

$\frac{- (\cos (2 x) - x) - \frac{- \sin (4 x)}{4}}{2} + C$

Step 21.5

Factor $- 1$ out of $- (\cos (2 x) - x) - \frac{- \sin (4 x)}{4}$ .

$\frac{- (\cos (2 x) - x + \frac{- \sin (4 x)}{4})}{2} + C$

Step 21.6

Rewrite $- (\cos (2 x) - x + \frac{- \sin (4 x)}{4})$ as $- 1 (\cos (2 x) - x + \frac{- \sin (4 x)}{4})$ .

$\frac{- 1 (\cos (2 x) - x + \frac{- \sin (4 x)}{4})}{2} + C$

Step 21.7

Move the negative in front of the fraction.

$- \frac{\cos (2 x) - x + \frac{- \sin (4 x)}{4}}{2} + C$

Step 21.8

Move the negative in front of the fraction.

$- \frac{\cos (2 x) - x - \frac{\sin (4 x)}{4}}{2} + C$

Step 21.9

Reorder terms.

$- \frac{1}{2} (\cos (2 x) - x - \frac{1}{4} \sin (4 x)) + C$