Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

Multiply both sides by $\frac{1}{e^{y}}$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Cancel the common factor of $e^{y}$ .

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Step 1.3.1.1

Factor $e^{y}$ out of $e^{y} \frac{\sin (x)}{\cos^{2} (x)}$ .

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

Step 1.3.1.2

Cancel the common factor.

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

Step 1.3.1.3

Rewrite the expression.

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

Step 1.3.2

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

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{\sin (x)}{\cos (x) \cos (x)}$

Step 1.3.3

Separate fractions.

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

Step 1.3.4

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

$\frac{1}{e^{y}} \frac{d y}{d x} = \frac{1}{\cos (x)} \tan (x)$

Step 1.3.5

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

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Negate the exponent of $e^{y}$ and move it out of the denominator.

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

Step 2.2.1.2

Simplify.

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Step 2.2.1.2.1

Multiply the exponents in ${(e^{y})}^{- 1}$ .

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Step 2.2.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int 1 e^{y \cdot - 1} d y = \int \sec (x) \tan (x) d x$

Step 2.2.1.2.1.2

Move $- 1$ to the left of $y$ .

$\int 1 e^{- 1 \cdot y} d y = \int \sec (x) \tan (x) d x$

Step 2.2.1.2.1.3

Rewrite $- 1 y$ as $- y$ .

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

Step 2.2.1.2.2

Multiply $e^{- y}$ by $1$ .

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

Step 2.2.2

Let $u = - y$ . Then $d u = - d y$ , so $- d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = - y$ . Find $\frac{d u}{d y}$ .

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Step 2.2.2.1.1

Differentiate $- y$ .

$\frac{d}{d y} [- y]$

Step 2.2.2.1.2

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

$- \frac{d}{d y} [y]$

Step 2.2.2.1.3

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

$- 1 \cdot 1$

Step 2.2.2.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.2.2.2

Rewrite the problem using $u$ and $d u$ .

$\int - e^{u} d u = \int \sec (x) \tan (x) d x$

Step 2.2.3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int e^{u} d u = \int \sec (x) \tan (x) d x$

Step 2.2.4

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$- (e^{u} + C_{1}) = \int \sec (x) \tan (x) d x$

Step 2.2.5

Simplify.

$- e^{u} + C_{1} = \int \sec (x) \tan (x) d x$

Step 2.2.6

Replace all occurrences of $u$ with $- y$ .

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

Step 2.3

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

$- e^{- y} + C_{1} = \sec (x) + C_{2}$

Step 2.4

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

$- e^{- y} = \sec (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $- e^{- y} = \sec (x) + K$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- e^{- y} = \sec (x) + K$ by $- 1$ .

$\frac{- e^{- y}}{- 1} = \frac{\sec (x)}{- 1} + \frac{K}{- 1}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

$\frac{e^{- y}}{1} = \frac{\sec (x)}{- 1} + \frac{K}{- 1}$

Step 3.1.2.2

Divide $e^{- y}$ by $1$ .

$e^{- y} = \frac{\sec (x)}{- 1} + \frac{K}{- 1}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move the negative one from the denominator of $\frac{\sec (x)}{- 1}$ .

$e^{- y} = - 1 \cdot \sec (x) + \frac{K}{- 1}$

Step 3.1.3.1.2

Rewrite $- 1 \cdot \sec (x)$ as $- \sec (x)$ .

$e^{- y} = - \sec (x) + \frac{K}{- 1}$

Step 3.1.3.1.3

Move the negative one from the denominator of $\frac{K}{- 1}$ .

$e^{- y} = - \sec (x) - 1 \cdot K$

Step 3.1.3.1.4

Rewrite $- 1 \cdot K$ as $- K$ .

$e^{- y} = - \sec (x) - K$

Step 3.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- y}) = \ln (- \sec (x) - K)$

Step 3.3

Expand the left side.

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Step 3.3.1

Expand $\ln (e^{- y})$ by moving $- y$ outside the logarithm.

$- y \ln (e) = \ln (- \sec (x) - K)$

Step 3.3.2

The natural logarithm of $e$ is $1$ .

$- y \cdot 1 = \ln (- \sec (x) - K)$

Step 3.3.3

Multiply $- 1$ by $1$ .

$- y = \ln (- \sec (x) - K)$

Step 3.4

Divide each term in $- y = \ln (- \sec (x) - K)$ by $- 1$ and simplify.

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Step 3.4.1

Divide each term in $- y = \ln (- \sec (x) - K)$ by $- 1$ .

$\frac{- y}{- 1} = \frac{\ln (- \sec (x) - K)}{- 1}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{\ln (- \sec (x) - K)}{- 1}$

Step 3.4.2.2

Divide $y$ by $1$ .

$y = \frac{\ln (- \sec (x) - K)}{- 1}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Move the negative one from the denominator of $\frac{\ln (- \sec (x) - K)}{- 1}$ .

$y = - 1 \cdot \ln (- \sec (x) - K)$

Step 3.4.3.2

Rewrite $- 1 \cdot \ln (- \sec (x) - K)$ as $- \ln (- \sec (x) - K)$ .

$y = - \ln (- \sec (x) - K)$

Step 4

Simplify the constant of integration.

$y = - \ln (- \sec (x) + K)$