Calculus Examples

$y^{- 1} d y + y e^{\cos (x)} \sin (x) d x = 0$

Step 1

Subtract $y e^{\cos (x)} \sin (x) d x$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

Multiply $\frac{1}{y} \cdot \frac{1}{y}$ .

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Raise $y$ to the power of $1$ .

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

Step 3.2.3

Raise $y$ to the power of $1$ .

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

Step 3.2.4

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

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

Step 3.2.5

Add $1$ and $1$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.4.1

Move the leading negative in $- \frac{1}{y}$ into the numerator.

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

Step 3.4.2

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

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

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

Step 4

Integrate both sides.

Tap for more steps...

Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

Tap for more steps...

Step 4.2.1

Apply basic rules of exponents.

Tap for more steps...

Step 4.2.1.1

Move $y^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.2.1.2

Multiply the exponents in ${(y^{2})}^{- 1}$ .

Tap for more steps...

Step 4.2.1.2.1

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

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

Step 4.2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 4.2.2

By the Power Rule, the integral of $y^{- 2}$ with respect to $y$ is $- y^{- 1}$ .

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

Step 4.2.3

Rewrite $- y^{- 1} + C_{1}$ as $- \frac{1}{y} + C_{1}$ .

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

Step 4.3

Integrate the right side.

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

Let $u = \cos (x)$ . Then $d u = - \sin (x) d x$ , so $- \frac{1}{\sin (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.3.2.1

Let $u = \cos (x)$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.3.2.1.1

Differentiate $\cos (x)$ .

$\frac{d}{d x} [\cos (x)]$

Step 4.3.2.1.2

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

$- \sin (x)$

Step 4.3.2.2

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

$- \frac{1}{y} + C_{1} = - \int - e^{u} d u$

Step 4.3.3

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

$- \frac{1}{y} + C_{1} = - - \int e^{u} d u$

Step 4.3.4

Simplify.

Tap for more steps...

Step 4.3.4.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{y} + C_{1} = 1 \int e^{u} d u$

Step 4.3.4.2

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

$- \frac{1}{y} + C_{1} = \int e^{u} d u$

Step 4.3.5

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

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

Step 4.3.6

Replace all occurrences of $u$ with $\cos (x)$ .

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

Step 4.4

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

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

Step 5

Solve for $y$ .

Tap for more steps...

Step 5.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 5.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y, 1, 1$

Step 5.1.2

The LCM of one and any expression is the expression.

$y$

Step 5.2

Multiply each term in $- \frac{1}{y} = e^{\cos (x)} + K$ by $y$ to eliminate the fractions.

Tap for more steps...

Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 5.2.2.1.1

Move the leading negative in $- \frac{1}{y}$ into the numerator.

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

Step 5.2.2.1.2

Cancel the common factor.

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

Step 5.2.2.1.3

Rewrite the expression.

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

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Reorder factors in $e^{\cos (x)} y + K y$ .

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

Step 5.3

Solve the equation.

Tap for more steps...

Step 5.3.1

Rewrite the equation as $y e^{\cos (x)} + K y = - 1$ .

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

Step 5.3.2

Factor $y$ out of $y e^{\cos (x)} + K y$ .

Tap for more steps...

Step 5.3.2.1

Factor $y$ out of $K y$ .

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

Step 5.3.2.2

Factor $y$ out of $y e^{\cos (x)} + y K$ .

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

Step 5.3.3

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

Tap for more steps...

Step 5.3.3.1

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

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

Step 5.3.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.3.2.1

Cancel the common factor of $e^{\cos (x)} + K$ .

Tap for more steps...

Step 5.3.3.2.1.1

Cancel the common factor.

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

Step 5.3.3.2.1.2

Divide $y$ by $1$ .

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

Step 5.3.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.3.1

Move the negative in front of the fraction.

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