Calculus Examples

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

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 $y \sec (x)$ .

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

Step 1.2

Reorder $y$ and $\sec (x)$ .

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

Step 2

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

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Step 2.1

Set up the integration.

$e^{\int \sec (x) d x}$

Step 2.2

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

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

Step 2.3

Remove the constant of integration.

$e^{\ln (\sec (x) + \tan (x))}$

Step 2.4

Exponentiation and log are inverse functions.

$\sec (x) + \tan (x)$

Step 3

Multiply each term by the integrating factor $\sec (x) + \tan (x)$ .

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Step 3.1

Multiply each term by $\sec (x) + \tan (x)$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.3.4

Add $1$ and $1$ .

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

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Multiply $\tan (x) \tan (x)$ .

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

Add $1$ and $1$ .

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

Step 3.5

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

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

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

Step 7.3

Using the Pythagorean Identity, rewrite $\tan^{2} (x)$ as $- 1 + \sec^{2} (x)$ .

$(\sec (x) + \tan (x)) y = \sec (x) + C + \int - 1 + \sec^{2} (x) d x$

Step 7.4

Split the single integral into multiple integrals.

$(\sec (x) + \tan (x)) y = \sec (x) + C + \int - 1 d x + \int \sec^{2} (x) d x$

Step 7.5

Apply the constant rule.

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

Step 7.6

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

$(\sec (x) + \tan (x)) y = \sec (x) + C - x + C + \tan (x) + C$

Step 7.7

Simplify.

$(\sec (x) + \tan (x)) y = \sec (x) - x + \tan (x) + C$

Step 8

Divide each term in $(\sec (x) + \tan (x)) y = \sec (x) - x + \tan (x) + C$ by $\sec (x) + \tan (x)$ and simplify.

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Step 8.1

Divide each term in $(\sec (x) + \tan (x)) y = \sec (x) - x + \tan (x) + C$ by $\sec (x) + \tan (x)$ .

$\frac{(\sec (x) + \tan (x)) y}{\sec (x) + \tan (x)} = \frac{\sec (x)}{\sec (x) + \tan (x)} + \frac{- x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\sec (x) + \tan (x)$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{(\sec (x) + \tan (x)) y}{\sec (x) + \tan (x)} = \frac{\sec (x)}{\sec (x) + \tan (x)} + \frac{- x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Step 8.2.1.2

Divide $y$ by $1$ .

$y = \frac{\sec (x)}{\sec (x) + \tan (x)} + \frac{- x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

$y = \frac{\sec (x)}{\sec (x) + \tan (x)} - \frac{x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Step 8.3.2

Combine the numerators over the common denominator.

$y = \frac{\sec (x) - x}{\sec (x) + \tan (x)} + \frac{\tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Step 8.3.3

Combine the numerators over the common denominator.

$y = \frac{\sec (x) - x + \tan (x)}{\sec (x) + \tan (x)} + \frac{C}{\sec (x) + \tan (x)}$

Step 8.3.4

Combine the numerators over the common denominator.

$y = \frac{\sec (x) - x + \tan (x) + C}{\sec (x) + \tan (x)}$