Calculus Examples

$\frac{d y}{d θ} = \frac{e^{y} \sin^{2} (θ)}{y \sec (θ)}$

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

$\frac{d y}{d θ} = \frac{e^{y}}{y} \cdot \frac{\sin^{2} (θ)}{\sec (θ)}$

Step 1.2

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

$\frac{y}{e^{y}} \frac{d y}{d θ} = \frac{y}{e^{y}} (\frac{e^{y}}{y} \cdot \frac{\sin^{2} (θ)}{\sec (θ)})$

Step 1.3

Simplify.

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Step 1.3.1

Combine.

$\frac{y}{e^{y}} \frac{d y}{d θ} = \frac{y}{e^{y}} \cdot \frac{e^{y} \sin^{2} (θ)}{y \sec (θ)}$

Step 1.3.2

Combine.

$\frac{y}{e^{y}} \frac{d y}{d θ} = \frac{y (e^{y} \sin^{2} (θ))}{e^{y} (y \sec (θ))}$

Step 1.3.3

Cancel the common factor of $y$ .

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Step 1.3.3.1

Cancel the common factor.

$\frac{y}{e^{y}} \frac{d y}{d θ} = \frac{y (e^{y} \sin^{2} (θ))}{e^{y} (y \sec (θ))}$

Step 1.3.3.2

Rewrite the expression.

$\frac{y}{e^{y}} \frac{d y}{d θ} = \frac{e^{y} \sin^{2} (θ)}{e^{y} (\sec (θ))}$

Step 1.3.4

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

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Step 1.3.4.1

Cancel the common factor.

$\frac{y}{e^{y}} \frac{d y}{d θ} = \frac{e^{y} \sin^{2} (θ)}{e^{y} \sec (θ)}$

Step 1.3.4.2

Rewrite the expression.

$\frac{y}{e^{y}} \frac{d y}{d θ} = \frac{\sin^{2} (θ)}{\sec (θ)}$

Step 1.3.5

Factor $\sin (θ)$ out of $\sin^{2} (θ)$ .

$\frac{y}{e^{y}} \frac{d y}{d θ} = \frac{\sin (θ) \sin (θ)}{\sec (θ)}$

Step 1.3.6

Separate fractions.

$\frac{y}{e^{y}} \frac{d y}{d θ} = \frac{\sin (θ)}{1} \cdot \frac{\sin (θ)}{\sec (θ)}$

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 1.3.9

Divide $\sin (θ)$ by $1$ .

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

Step 1.3.10

Multiply $\sin (θ) (\sin (θ) \cos (θ))$ .

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Step 1.3.10.1

Raise $\sin (θ)$ to the power of $1$ .

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

Step 1.3.10.2

Raise $\sin (θ)$ to the power of $1$ .

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

Step 1.3.10.3

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

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

Step 1.3.10.4

Add $1$ and $1$ .

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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.

$\int y {(e^{y})}^{- 1} d y = \int \sin^{2} (θ) \cos (θ) d θ$

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

$\int y e^{y \cdot - 1} d y = \int \sin^{2} (θ) \cos (θ) d θ$

Step 2.2.1.2.2

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

$\int y e^{- 1 \cdot y} d y = \int \sin^{2} (θ) \cos (θ) d θ$

Step 2.2.1.2.3

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

$\int y e^{- y} d y = \int \sin^{2} (θ) \cos (θ) d θ$

Step 2.2.2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = y$ and $d v = e^{- y}$ .

$y (- e^{- y}) - \int - e^{- y} d y = \int \sin^{2} (θ) \cos (θ) d θ$

Step 2.2.3

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

$y (- e^{- y}) - - \int e^{- y} d y = \int \sin^{2} (θ) \cos (θ) d θ$

Step 2.2.4

Simplify.

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Step 2.2.4.1

Multiply $- 1$ by $- 1$ .

$y (- e^{- y}) + 1 \int e^{- y} d y = \int \sin^{2} (θ) \cos (θ) d θ$

Step 2.2.4.2

Multiply $\int e^{- y} d y$ by $1$ .

$y (- e^{- y}) + \int e^{- y} d y = \int \sin^{2} (θ) \cos (θ) d θ$

Step 2.2.5

Let $u_{1} = - y$ . Then $d u_{1} = - d y$ , so $- d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.5.1

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

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Step 2.2.5.1.1

Differentiate $- y$ .

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

Step 2.2.5.1.2

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 2.2.5.1.3

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 2.2.5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 2.2.5.2

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

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

Step 2.2.6

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

$y (- e^{- y}) - \int e^{u_{1}} d u_{1} = \int \sin^{2} (θ) \cos (θ) d θ$

Step 2.2.7

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

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

Step 2.2.8

Rewrite $y (- e^{- y}) - (e^{u_{1}} + C_{1})$ as $- y e^{- y} - e^{u_{1}} + C_{1}$ .

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

Step 2.2.9

Replace all occurrences of $u_{1}$ with $- y$ .

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

Step 2.2.10

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u_{2} = \sin (θ)$ . Then $d u_{2} = \cos (θ) d θ$ , so $\frac{1}{\cos (θ)} d u_{2} = d θ$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.1.1

Let $u_{2} = \sin (θ)$ . Find $\frac{d u_{2}}{d θ}$ .

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Step 2.3.1.1.1

Differentiate $\sin (θ)$ .

$\frac{d}{d θ} [\sin (θ)]$

Step 2.3.1.1.2

The derivative of $\sin (θ)$ with respect to $θ$ is $\cos (θ)$ .

$\cos (θ)$

Step 2.3.1.2

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

$- e^{- y} y - e^{- y} + C_{1} = \int {u_{2}}^{2} d u_{2}$

Step 2.3.2

By the Power Rule, the integral of ${u_{2}}^{2}$ with respect to $u_{2}$ is $\frac{1}{3} {u_{2}}^{3}$ .

$- e^{- y} y - e^{- y} + C_{1} = \frac{1}{3} {u_{2}}^{3} + C_{2}$

Step 2.3.3

Replace all occurrences of $u_{2}$ with $\sin (θ)$ .

$- e^{- y} y - e^{- y} + C_{1} = \frac{1}{3} \sin^{3} (θ) + C_{2}$

Step 2.4

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

$- e^{- y} y - e^{- y} = \frac{1}{3} \sin^{3} (θ) + K$