Calculus Examples

$\frac{d y}{d x} + 2 y \tan (x) = \sin (x)$

Step 1

Rewrite the differential equation as $\frac{d y}{d x} + M (x) y = Q (x)$ .

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Step 1.1

Factor $y$ out of $2 y \tan (x)$ .

$\frac{d y}{d x} + y (2 \tan (x)) = \sin (x)$

Step 1.2

Reorder $y$ and $2 \tan (x)$ .

$\frac{d y}{d x} + 2 \tan (x) y = \sin (x)$

Step 2

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 2 \tan (x)$ .

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Step 2.1

Set up the integration.

$e^{\int 2 \tan (x) d x}$

Step 2.2

Integrate $2 \tan (x)$ .

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Step 2.2.1

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

$e^{2 \int \tan (x) d x}$

Step 2.2.2

The integral of $\tan (x)$ with respect to $x$ is $\ln (| \sec (x) |)$ .

$e^{2 (\ln (| \sec (x) |) + C)}$

Step 2.2.3

Simplify.

$e^{2 \ln (| \sec (x) |) + C}$

Step 2.3

Remove the constant of integration.

$e^{2 \ln (\sec (x))}$

Step 2.4

Use the logarithmic power rule.

$e^{\ln (\sec^{2} (x))}$

Step 2.5

Exponentiation and log are inverse functions.

$\sec^{2} (x)$

Step 3

Multiply each term by the integrating factor $\sec^{2} (x)$ .

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Step 3.1

Multiply each term by $\sec^{2} (x)$ .

$\sec^{2} (x) \frac{d y}{d x} + \sec^{2} (x) (2 \tan (x) y) = \sec^{2} (x) \sin (x)$

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite $\sec (x)$ in terms of sines and cosines.

${(\frac{1}{\cos (x)})}^{2} \frac{d y}{d x} + \sec^{2} (x) (2 \tan (x) y) = \sec^{2} (x) \sin (x)$

Step 3.2.2

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

$\frac{1^{2}}{\cos^{2} (x)} \frac{d y}{d x} + \sec^{2} (x) (2 \tan (x) y) = \sec^{2} (x) \sin (x)$

Step 3.2.3

One to any power is one.

$\frac{1}{\cos^{2} (x)} \frac{d y}{d x} + \sec^{2} (x) (2 \tan (x) y) = \sec^{2} (x) \sin (x)$

Step 3.2.4

Combine $\frac{1}{\cos^{2} (x)}$ and $\frac{d y}{d x}$ .

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \sec^{2} (x) (2 \tan (x) y) = \sec^{2} (x) \sin (x)$

Step 3.2.5

Rewrite using the commutative property of multiplication.

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + 2 \sec^{2} (x) \tan (x) y = \sec^{2} (x) \sin (x)$

Step 3.2.6

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + 2 {(\frac{1}{\cos (x)})}^{2} \tan (x) y = \sec^{2} (x) \sin (x)$

Step 3.2.7

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

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + 2 \frac{1^{2}}{\cos^{2} (x)} \tan (x) y = \sec^{2} (x) \sin (x)$

Step 3.2.8

One to any power is one.

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + 2 \frac{1}{\cos^{2} (x)} \tan (x) y = \sec^{2} (x) \sin (x)$

Step 3.2.9

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

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \frac{2}{\cos^{2} (x)} \tan (x) y = \sec^{2} (x) \sin (x)$

Step 3.2.10

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \frac{2}{\cos^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)} y = \sec^{2} (x) \sin (x)$

Step 3.2.11

Combine.

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \frac{2 \sin (x)}{\cos^{2} (x) \cos (x)} y = \sec^{2} (x) \sin (x)$

Step 3.2.12

Multiply $\cos^{2} (x)$ by $\cos (x)$ by adding the exponents.

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Step 3.2.12.1

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 3.2.12.1.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.2.12.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.12.2

Add $2$ and $1$ .

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \frac{2 \sin (x)}{\cos^{3} (x)} y = \sec^{2} (x) \sin (x)$

Step 3.2.13

Combine $\frac{2 \sin (x)}{\cos^{3} (x)}$ and $y$ .

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \frac{2 \sin (x) y}{\cos^{3} (x)} = \sec^{2} (x) \sin (x)$

Step 3.3

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \frac{2 \sin (x) y}{\cos^{3} (x)} = {(\frac{1}{\cos (x)})}^{2} \sin (x)$

Step 3.4

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

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \frac{2 \sin (x) y}{\cos^{3} (x)} = \frac{1^{2}}{\cos^{2} (x)} \sin (x)$

Step 3.5

One to any power is one.

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \frac{2 \sin (x) y}{\cos^{3} (x)} = \frac{1}{\cos^{2} (x)} \sin (x)$

Step 3.6

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

$\frac{\frac{d y}{d x}}{\cos^{2} (x)} + \frac{2 \sin (x) y}{\cos^{3} (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7

Simplify each term.

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Step 3.7.1

Multiply by $1$ .

$\frac{\frac{d y}{d x}}{\cos^{2} (x) \cdot 1} + \frac{2 \sin (x) y}{\cos^{3} (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.2

Separate fractions.

$\frac{\frac{d y}{d x}}{1} \cdot \frac{1}{\cos^{2} (x)} + \frac{2 \sin (x) y}{\cos^{3} (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.3

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

$\frac{\frac{d y}{d x}}{1} \sec^{2} (x) + \frac{2 \sin (x) y}{\cos^{3} (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.4

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

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 \sin (x) y}{\cos^{3} (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.5

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 \sin (x) y}{\cos (x) \cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.6

Separate fractions.

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 y}{\cos^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.7

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 y}{\cos^{2} (x)} \tan (x) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.8

Combine $\frac{2 y}{\cos^{2} (x)}$ and $\tan (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 y \tan (x)}{\cos^{2} (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.9

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 y \tan (x)}{\cos (x) \cos (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.10

Separate fractions.

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 y}{\cos (x)} \cdot \frac{\tan (x)}{\cos (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.11

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 y}{\cos (x)} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.12

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ as a product.

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 y}{\cos (x)} (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.13

Simplify.

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Step 3.7.13.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 y}{\cos (x)} (\tan (x) \frac{1}{\cos (x)}) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.13.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{2 y}{\cos (x)} (\tan (x) \sec (x)) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.14

Multiply $\frac{2 y}{\cos (x)} (\tan (x) \sec (x))$ .

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Step 3.7.14.1

Combine $\tan (x)$ and $\frac{2 y}{\cos (x)}$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{\tan (x) (2 y)}{\cos (x)} \sec (x) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.14.2

Combine $\frac{\tan (x) (2 y)}{\cos (x)}$ and $\sec (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{\tan (x) (2 y) \sec (x)}{\cos (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.15

Separate fractions.

$\frac{d y}{d x} \sec^{2} (x) + \frac{(2 y) \sec (x)}{1} \cdot \frac{\tan (x)}{\cos (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.16

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{d y}{d x} \sec^{2} (x) + \frac{(2 y) \sec (x)}{1} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.17

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ as a product.

$\frac{d y}{d x} \sec^{2} (x) + \frac{(2 y) \sec (x)}{1} (\frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.18

Simplify.

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Step 3.7.18.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{(2 y) \sec (x)}{1} (\tan (x) \frac{1}{\cos (x)}) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.18.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + \frac{(2 y) \sec (x)}{1} (\tan (x) \sec (x)) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.19

Divide $(2 y) \sec (x)$ by $1$ .

$\frac{d y}{d x} \sec^{2} (x) + (2 y) \sec (x) (\tan (x) \sec (x)) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.20

Multiply $2 y \sec (x) (\tan (x) \sec (x))$ .

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Step 3.7.20.1

Raise $\sec (x)$ to the power of $1$ .

$\frac{d y}{d x} \sec^{2} (x) + 2 y (\sec^{1} (x) \sec (x)) \tan (x) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.20.2

Raise $\sec (x)$ to the power of $1$ .

$\frac{d y}{d x} \sec^{2} (x) + 2 y (\sec^{1} (x) \sec^{1} (x)) \tan (x) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.20.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d y}{d x} \sec^{2} (x) + 2 y {\sec (x)}^{1 + 1} \tan (x) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.7.20.4

Add $1$ and $1$ .

$\frac{d y}{d x} \sec^{2} (x) + 2 y \sec^{2} (x) \tan (x) = \frac{\sin (x)}{\cos^{2} (x)}$

Step 3.8

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + 2 y \sec^{2} (x) \tan (x) = \frac{\sin (x)}{\cos (x) \cos (x)}$

Step 3.9

Separate fractions.

$\frac{d y}{d x} \sec^{2} (x) + 2 y \sec^{2} (x) \tan (x) = \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$

Step 3.10

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

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

Step 3.11

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{d y}{d x} \sec^{2} (x) + 2 y \sec^{2} (x) \tan (x) = \sec (x) \tan (x)$

Step 3.12

Reorder factors in $\frac{d y}{d x} \sec^{2} (x) + 2 y \sec^{2} (x) \tan (x) = \sec (x) \tan (x)$ .

$\sec^{2} (x) \frac{d y}{d x} + 2 y \sec^{2} (x) \tan (x) = \sec (x) \tan (x)$

Step 4

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [\sec^{2} (x) y] = \sec (x) \tan (x)$

Step 5

Set up an integral on each side.

$\int \frac{d}{d x} [\sec^{2} (x) y] d x = \int \sec (x) \tan (x) d x$

Step 6

Integrate the left side.

$\sec^{2} (x) y = \int \sec (x) \tan (x) d x$

Step 7

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

$\sec^{2} (x) y = \sec (x) + C$

Step 8

Divide each term in $\sec^{2} (x) y = \sec (x) + C$ by $\sec^{2} (x)$ and simplify.

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Step 8.1

Divide each term in $\sec^{2} (x) y = \sec (x) + C$ by $\sec^{2} (x)$ .

$\frac{\sec^{2} (x) y}{\sec^{2} (x)} = \frac{\sec (x)}{\sec^{2} (x)} + \frac{C}{\sec^{2} (x)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\sec^{2} (x)$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{\sec^{2} (x) y}{\sec^{2} (x)} = \frac{\sec (x)}{\sec^{2} (x)} + \frac{C}{\sec^{2} (x)}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\sec (x)}{\sec^{2} (x)} + \frac{C}{\sec^{2} (x)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Cancel the common factor of $\sec (x)$ and $\sec^{2} (x)$ .

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Step 8.3.1.1.1

Multiply by $1$ .

$y = \frac{\sec (x) \cdot 1}{\sec^{2} (x)} + \frac{C}{\sec^{2} (x)}$

Step 8.3.1.1.2

Cancel the common factors.

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Step 8.3.1.1.2.1

Factor $\sec (x)$ out of $\sec^{2} (x)$ .

$y = \frac{\sec (x) \cdot 1}{\sec (x) \sec (x)} + \frac{C}{\sec^{2} (x)}$

Step 8.3.1.1.2.2

Cancel the common factor.

$y = \frac{\sec (x) \cdot 1}{\sec (x) \sec (x)} + \frac{C}{\sec^{2} (x)}$

Step 8.3.1.1.2.3

Rewrite the expression.

$y = \frac{1}{\sec (x)} + \frac{C}{\sec^{2} (x)}$

Step 8.3.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$y = \frac{1}{\frac{1}{\cos (x)}} + \frac{C}{\sec^{2} (x)}$

Step 8.3.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$y = 1 \cos (x) + \frac{C}{\sec^{2} (x)}$

Step 8.3.1.4

Multiply $\cos (x)$ by $1$ .

$y = \cos (x) + \frac{C}{\sec^{2} (x)}$