Calculus Examples

$(1 + \cos (2 x)) \frac{d y}{d x} = 2$ , $y (\frac{π}{4}) = 1$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $(1 + \cos (2 x)) \frac{d y}{d x} = 2$ by $1 + \cos (2 x)$ and simplify.

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Step 1.1.1

Divide each term in $(1 + \cos (2 x)) \frac{d y}{d x} = 2$ by $1 + \cos (2 x)$ .

$\frac{(1 + \cos (2 x)) \frac{d y}{d x}}{1 + \cos (2 x)} = \frac{2}{1 + \cos (2 x)}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $1 + \cos (2 x)$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{(1 + \cos (2 x)) \frac{d y}{d x}}{1 + \cos (2 x)} = \frac{2}{1 + \cos (2 x)}$

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{2}{1 + \cos (2 x)}$

Step 1.2

Rewrite the equation.

$d y = \frac{2}{1 + \cos (2 x)} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int d y = \int \frac{2}{1 + \cos (2 x)} d x$

Step 2.2

Apply the constant rule.

$y + C_{1} = \int \frac{2}{1 + \cos (2 x)} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$y + C_{1} = 2 \int \frac{1}{1 + \cos (2 x)} d x$

Step 2.3.2

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 2.3.2.1

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

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Step 2.3.2.1.1

Differentiate $2 x$ .

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

Step 2.3.2.1.2

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 2.3.2.1.3

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 2.3.2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.3.2.2

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

$y + C_{1} = 2 \int \frac{1}{1 + \cos (u_{1})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Multiply $\frac{1}{1 + \cos (u_{1})}$ by $\frac{1}{2}$ .

$y + C_{1} = 2 \int \frac{1}{(1 + \cos (u_{1})) \cdot 2} d u_{1}$

Step 2.3.3.2

Move $2$ to the left of $1 + \cos (u_{1})$ .

$y + C_{1} = 2 \int \frac{1}{2 (1 + \cos (u_{1}))} d u_{1}$

Step 2.3.4

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

$y + C_{1} = 2 (\frac{1}{2} \int \frac{1}{1 + \cos (u_{1})} d u_{1})$

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

$y + C_{1} = \frac{2}{2} \int \frac{1}{1 + \cos (u_{1})} d u_{1}$

Step 2.3.5.2

Cancel the common factor of $2$ .

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Step 2.3.5.2.1

Cancel the common factor.

$y + C_{1} = \frac{2}{2} \int \frac{1}{1 + \cos (u_{1})} d u_{1}$

Step 2.3.5.2.2

Rewrite the expression.

$y + C_{1} = 1 \int \frac{1}{1 + \cos (u_{1})} d u_{1}$

Step 2.3.5.3

Multiply $\int \frac{1}{1 + \cos (u_{1})} d u_{1}$ by $1$ .

$y + C_{1} = \int \frac{1}{1 + \cos (u_{1})} d u_{1}$

Step 2.3.6

Use the double-angle identity to transform $\cos (x)$ to $\cos^{2} (\frac{x}{2}) - \sin^{2} (\frac{x}{2})$ .

$y + C_{1} = \int \frac{1}{1 + \cos^{2} (\frac{u_{1}}{2}) - \sin^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.7

Use the pythagorean identity to transform $1$ to $\sin^{2} (\frac{u_{1}}{2}) + \cos^{2} (\frac{u_{1}}{2})$ .

$y + C_{1} = \int \frac{1}{\sin^{2} (\frac{u_{1}}{2}) + \cos^{2} (\frac{u_{1}}{2}) + \cos^{2} (\frac{u_{1}}{2}) - \sin^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.8

Simplify.

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Step 2.3.8.1

Subtract ${\sin (\frac{u_{1}}{2})}^{2}$ from ${\sin (\frac{u_{1}}{2})}^{2}$ .

$y + C_{1} = \int \frac{1}{{\cos (\frac{u_{1}}{2})}^{2} + {\cos (\frac{u_{1}}{2})}^{2} + 0} d u_{1}$

Step 2.3.8.2

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

$y + C_{1} = \int \frac{1}{2 {\cos (\frac{u_{1}}{2})}^{2} + 0} d u_{1}$

Step 2.3.8.3

Add $2 {\cos (\frac{u_{1}}{2})}^{2}$ and $0$ .

$y + C_{1} = \int \frac{1}{2 \cos^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.9

Multiply the argument by $\frac{\sec^{2} (\frac{u}{2})}{\sec^{2} (\frac{u}{2})}$

$y + C_{1} = \int \frac{1}{2 \cos^{2} (\frac{u_{1}}{2})} \cdot \frac{\sec^{2} (\frac{u}{2})}{\sec^{2} (\frac{u}{2})} d u_{1}$

Step 2.3.10

Combine.

$y + C_{1} = \int \frac{1 \sec^{2} (\frac{u}{2})}{2 \cos^{2} (\frac{u_{1}}{2}) \sec^{2} (\frac{u}{2})} d u_{1}$

Step 2.3.11

Multiply ${\sec (\frac{u}{2})}^{2}$ by $1$ .

$y + C_{1} = \int \frac{\sec^{2} (\frac{u}{2})}{2 \cos^{2} (\frac{u_{1}}{2}) \sec^{2} (\frac{u}{2})} d u_{1}$

Step 2.3.12

Rewrite $\sec (\frac{u}{2})$ in terms of sines and cosines.

$y + C_{1} = \int \frac{\sec^{2} (\frac{u}{2})}{2 \cos^{2} (\frac{u_{1}}{2}) {(\frac{1}{\cos (\frac{u}{2})})}^{2}} d u_{1}$

Step 2.3.13

Apply the product rule to $\frac{1}{\cos (\frac{u}{2})}$ .

$y + C_{1} = \int \frac{\sec^{2} (\frac{u}{2})}{2 \cos^{2} (\frac{u_{1}}{2}) \frac{1^{2}}{\cos^{2} (\frac{u}{2})}} d u_{1}$

Step 2.3.14

One to any power is one.

$y + C_{1} = \int \frac{\sec^{2} (\frac{u}{2})}{2 \cos^{2} (\frac{u_{1}}{2}) \frac{1}{\cos^{2} (\frac{u}{2})}} d u_{1}$

Step 2.3.15

Multiply $2 \cos^{2} (\frac{u_{1}}{2}) \frac{1}{\cos^{2} (\frac{u}{2})}$ .

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Step 2.3.15.1

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

$y + C_{1} = \int \frac{\sec^{2} (\frac{u}{2})}{\frac{2}{\cos^{2} (\frac{u}{2})} \cos^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.15.2

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

$y + C_{1} = \int \frac{\sec^{2} (\frac{u}{2})}{\frac{2 \cos^{2} (\frac{u_{1}}{2})}{\cos^{2} (\frac{u}{2})}} d u_{1}$

Step 2.3.16

Multiply the numerator by the reciprocal of the denominator.

$y + C_{1} = \int \sec^{2} (\frac{u}{2}) \frac{\cos^{2} (\frac{u}{2})}{2 \cos^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.17

Rewrite $\sec (\frac{u}{2})$ in terms of sines and cosines.

$y + C_{1} = \int {(\frac{1}{\cos (\frac{u}{2})})}^{2} \frac{\cos^{2} (\frac{u}{2})}{2 \cos^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.18

Apply the product rule to $\frac{1}{\cos (\frac{u}{2})}$ .

$y + C_{1} = \int \frac{1^{2}}{\cos^{2} (\frac{u}{2})} \cdot \frac{\cos^{2} (\frac{u}{2})}{2 \cos^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.19

Combine.

$y + C_{1} = \int \frac{1^{2} \cos^{2} (\frac{u}{2})}{\cos^{2} (\frac{u}{2}) (2 \cos^{2} (\frac{u_{1}}{2}))} d u_{1}$

Step 2.3.20

Cancel the common factor of $\cos^{2} (\frac{u}{2})$ .

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Step 2.3.20.1

Cancel the common factor.

$y + C_{1} = \int \frac{1^{2} \cos^{2} (\frac{u}{2})}{\cos^{2} (\frac{u}{2}) (2 \cos^{2} (\frac{u_{1}}{2}))} d u_{1}$

Step 2.3.20.2

Rewrite the expression.

$y + C_{1} = \int \frac{1^{2}}{2 \cos^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.21

One to any power is one.

$y + C_{1} = \int \frac{1}{2 \cos^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.22

Multiply by $1$ .

$y + C_{1} = \int \frac{1}{2 (\cos^{2} (\frac{u_{1}}{2}) \cdot 1)} d u_{1}$

Step 2.3.23

Separate fractions.

$y + C_{1} = \int \frac{1}{2 \cdot (1)} \cdot \frac{1}{\cos^{2} (\frac{u_{1}}{2})} d u_{1}$

Step 2.3.24

Convert from $\frac{1}{\cos^{2} (\frac{u_{1}}{2})}$ to $\sec^{2} (\frac{u_{1}}{2})$ .

$y + C_{1} = \int \frac{1}{2 \cdot (1)} \sec^{2} (\frac{u_{1}}{2}) d u_{1}$

Step 2.3.25

Combine fractions.

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Step 2.3.25.1

Multiply $2$ by $1$ .

$y + C_{1} = \int \frac{1}{2} \sec^{2} (\frac{u_{1}}{2}) d u_{1}$

Step 2.3.25.2

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

$y + C_{1} = \int \frac{\sec^{2} (\frac{u_{1}}{2})}{2} d u_{1}$

Step 2.3.26

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

$y + C_{1} = \frac{1}{2} \int \sec^{2} (\frac{u_{1}}{2}) d u_{1}$

Step 2.3.27

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

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Step 2.3.27.1

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

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Step 2.3.27.1.1

Differentiate $\frac{u_{1}}{2}$ .

$\frac{d}{d u_{1}} [\frac{u_{1}}{2}]$

Step 2.3.27.1.2

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , the derivative of $\frac{u_{1}}{2}$ with respect to $u_{1}$ is $\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{1}{2} \frac{d}{d u_{1}} [u_{1}]$

Step 2.3.27.1.3

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

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

Step 2.3.27.1.4

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

$\frac{1}{2}$

Step 2.3.27.2

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

$y + C_{1} = \frac{1}{2} \int \sec^{2} (u_{2}) \frac{1}{\frac{1}{2}} d u_{2}$

Step 2.3.28

Simplify.

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Step 2.3.28.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$y + C_{1} = \frac{1}{2} \int \sec^{2} (u_{2}) (1 \cdot 2) d u_{2}$

Step 2.3.28.2

Multiply $2$ by $1$ .

$y + C_{1} = \frac{1}{2} \int \sec^{2} (u_{2}) \cdot 2 d u_{2}$

Step 2.3.28.3

Move $2$ to the left of $\sec^{2} (u_{2})$ .

$y + C_{1} = \frac{1}{2} \int 2 \sec^{2} (u_{2}) d u_{2}$

Step 2.3.29

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

$y + C_{1} = \frac{1}{2} (2 \int \sec^{2} (u_{2}) d u_{2})$

Step 2.3.30

Simplify.

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Step 2.3.30.1

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

$y + C_{1} = \frac{2}{2} \int \sec^{2} (u_{2}) d u_{2}$

Step 2.3.30.2

Cancel the common factor of $2$ .

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Step 2.3.30.2.1

Cancel the common factor.

$y + C_{1} = \frac{2}{2} \int \sec^{2} (u_{2}) d u_{2}$

Step 2.3.30.2.2

Rewrite the expression.

$y + C_{1} = 1 \int \sec^{2} (u_{2}) d u_{2}$

Step 2.3.30.3

Multiply $\int \sec^{2} (u_{2}) d u_{2}$ by $1$ .

$y + C_{1} = \int \sec^{2} (u_{2}) d u_{2}$

Step 2.3.31

Since the derivative of $\tan (u_{2})$ is $\sec^{2} (u_{2})$ , the integral of $\sec^{2} (u_{2})$ is $\tan (u_{2})$ .

$y + C_{1} = \tan (u_{2}) + C_{2}$

Step 2.3.32

Substitute back in for each integration substitution variable.

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Step 2.3.32.1

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

$y + C_{1} = \tan (\frac{u_{1}}{2}) + C_{2}$

Step 2.3.32.2

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

$y + C_{1} = \tan (\frac{2 x}{2}) + C_{2}$

Step 2.3.33

Simplify.

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Step 2.3.33.1

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

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Step 2.3.33.1.1

Cancel the common factor.

$y + C_{1} = \tan (\frac{2 x}{2}) + C_{2}$

Step 2.3.33.1.2

Rewrite the expression.

$y + C_{1} = \tan (\frac{x}{1}) + C_{2}$

Step 2.3.33.2

Divide $x$ by $1$ .

$y + C_{1} = \tan (x) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$y = \tan (x) + K$

Step 3

Use the initial condition to find the value of $K$ by substituting $\frac{π}{4}$ for $x$ and $1$ for $y$ in $y = \tan (x) + K$ .

$1 = \tan (\frac{π}{4}) + K$

Step 4

Solve for $K$ .

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Step 4.1

Rewrite the equation as $\tan (\frac{π}{4}) + K = 1$ .

$\tan (\frac{π}{4}) + K = 1$

Step 4.2

Simplify the left side.

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Step 4.2.1

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$1 + K = 1$

Step 4.3

Move all terms not containing $K$ to the right side of the equation.

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Step 4.3.1

Subtract $1$ from both sides of the equation.

$K = 1 - 1$

Step 4.3.2

Subtract $1$ from $1$ .

$K = 0$

Step 5

Substitute $0$ for $K$ in $y = \tan (x) + K$ and simplify.

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Step 5.1

Substitute $0$ for $K$ .

$y = \tan (x) + 0$

Step 5.2

Add $\tan (x)$ and $0$ .

$y = \tan (x)$