Calculus Examples

$(\sec (x)) \frac{d y}{d x} = e^{y + \sin (x)}$ , $y (0) = 0$

Step 1

Let $v = e^{y + \sin (x)}$ . Substitute $v$ for all occurrences of $e^{y + \sin (x)}$ .

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

Step 2

Find $\frac{d v}{d x}$ by differentiating $e^{y + \sin (x)}$ .

Tap for more steps...

Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = y + \sin (x)$ .

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $y + \sin (x)$ .

$\frac{d v}{d x} = \frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [y + \sin (x)]$

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$\frac{d v}{d x} = e^{u_{1}} \frac{d}{d x} [y + \sin (x)]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $y + \sin (x)$ .

$\frac{d v}{d x} = e^{y + \sin (x)} \frac{d}{d x} [y + \sin (x)]$

Step 2.2

By the Sum Rule, the derivative of $y + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [y] + \frac{d}{d x} [\sin (x)]$ .

$\frac{d v}{d x} = e^{y + \sin (x)} (\frac{d}{d x} [y] + \frac{d}{d x} [\sin (x)])$

Step 2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{d v}{d x} = e^{y + \sin (x)} (y' + \frac{d}{d x} [\sin (x)])$

Step 2.4

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

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

Step 3

Substitute $v$ for $e^{y + \sin (x)}$ .

$\frac{d v}{d x} = v (y' + \cos (x))$

Step 4

Substitute the derivative back in to the differential equation.

$\frac{\sec (x) \frac{d v}{d x}}{v} - 1 = v$

Step 5

Separate the variables.

Tap for more steps...

Step 5.1

Solve for $\frac{d v}{d x}$ .

Tap for more steps...

Step 5.1.1

Simplify the left side.

Tap for more steps...

Step 5.1.1.1

Reorder factors in $\frac{\sec (x) \frac{d v}{d x}}{v} - 1$ .

$\frac{\frac{d v}{d x} \sec (x)}{v} - 1 = v$

Step 5.1.2

Add $1$ to both sides of the equation.

$\frac{\frac{d v}{d x} \sec (x)}{v} = v + 1$

Step 5.1.3

Multiply both sides by $v$ .

$\frac{\frac{d v}{d x} \sec (x)}{v} v = (v + 1) v$

Step 5.1.4

Simplify.

Tap for more steps...

Step 5.1.4.1

Simplify the left side.

Tap for more steps...

Step 5.1.4.1.1

Cancel the common factor of $v$ .

Tap for more steps...

Step 5.1.4.1.1.1

Cancel the common factor.

$\frac{\frac{d v}{d x} \sec (x)}{v} v = (v + 1) v$

Step 5.1.4.1.1.2

Rewrite the expression.

$\frac{d v}{d x} \sec (x) = (v + 1) v$

Step 5.1.4.2

Simplify the right side.

Tap for more steps...

Step 5.1.4.2.1

Simplify $(v + 1) v$ .

Tap for more steps...

Step 5.1.4.2.1.1

Apply the distributive property.

$\frac{d v}{d x} \sec (x) = v \cdot v + 1 v$

Step 5.1.4.2.1.2

Simplify the expression.

Tap for more steps...

Step 5.1.4.2.1.2.1

Multiply $v$ by $v$ .

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

Step 5.1.4.2.1.2.2

Multiply $v$ by $1$ .

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

Step 5.1.5

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

Tap for more steps...

Step 5.1.5.1

Divide each term in $\frac{d v}{d x} \sec (x) = v^{2} + v$ by $\sec (x)$ .

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

Step 5.1.5.2

Simplify the left side.

Tap for more steps...

Step 5.1.5.2.1

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

Tap for more steps...

Step 5.1.5.2.1.1

Cancel the common factor.

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

Step 5.1.5.2.1.2

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

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

Step 5.1.5.3

Simplify the right side.

Tap for more steps...

Step 5.1.5.3.1

Simplify each term.

Tap for more steps...

Step 5.1.5.3.1.1

Separate fractions.

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

Step 5.1.5.3.1.2

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

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

Step 5.1.5.3.1.3

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

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

Step 5.1.5.3.1.4

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

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

Step 5.1.5.3.1.5

Divide $v^{2}$ by $1$ .

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

Step 5.1.5.3.1.6

Separate fractions.

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

Step 5.1.5.3.1.7

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

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

Step 5.1.5.3.1.8

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

$\frac{d v}{d x} = v^{2} \cos (x) + \frac{v}{1} (1 \cos (x))$

Step 5.1.5.3.1.9

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

$\frac{d v}{d x} = v^{2} \cos (x) + \frac{v}{1} \cos (x)$

Step 5.1.5.3.1.10

Divide $v$ by $1$ .

$\frac{d v}{d x} = v^{2} \cos (x) + v \cos (x)$

Step 5.2

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

Tap for more steps...

Step 5.2.1

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

$\frac{d v}{d x} = v \cos (x) (v) + v \cos (x)$

Step 5.2.2

Factor $v \cos (x)$ out of $v \cos (x)$ .

$\frac{d v}{d x} = v \cos (x) (v) + v \cos (x) (1)$

Step 5.2.3

Factor $v \cos (x)$ out of $v \cos (x) (v) + v \cos (x) (1)$ .

$\frac{d v}{d x} = v \cos (x) (v + 1)$

Step 5.3

Multiply both sides by $\frac{1}{v (v + 1)}$ .

$\frac{1}{v (v + 1)} \frac{d v}{d x} = \frac{1}{v (v + 1)} (v \cos (x) (v + 1))$

Step 5.4

Cancel the common factor of $v (v + 1)$ .

Tap for more steps...

Step 5.4.1

Factor $v (v + 1)$ out of $v \cos (x) (v + 1)$ .

$\frac{1}{v (v + 1)} \frac{d v}{d x} = \frac{1}{v (v + 1)} (v (v + 1) (\cos (x)))$

Step 5.4.2

Cancel the common factor.

$\frac{1}{v (v + 1)} \frac{d v}{d x} = \frac{1}{v (v + 1)} (v (v + 1) \cos (x))$

Step 5.4.3

Rewrite the expression.

$\frac{1}{v (v + 1)} \frac{d v}{d x} = \cos (x)$

Step 5.5

Rewrite the equation.

$\frac{1}{v (v + 1)} d v = \cos (x) d x$

Step 6

Integrate both sides.

Tap for more steps...

Step 6.1

Set up an integral on each side.

$\int \frac{1}{v (v + 1)} d v = \int \cos (x) d x$

Step 6.2

Integrate the left side.

Tap for more steps...

Step 6.2.1

Write the fraction using partial fraction decomposition.

Tap for more steps...

Step 6.2.1.1

Decompose the fraction and multiply through by the common denominator.

Tap for more steps...

Step 6.2.1.1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $B$ .

$\frac{A}{v} + \frac{B}{v + 1}$

Step 6.2.1.1.2

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $v (v + 1)$ .

$\frac{1 (v (v + 1))}{v (v + 1)} = \frac{A (v (v + 1))}{v} + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.3

Cancel the common factor of $v$ .

Tap for more steps...

Step 6.2.1.1.3.1

Cancel the common factor.

$\frac{1 (v (v + 1))}{v (v + 1)} = \frac{A (v (v + 1))}{v} + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.3.2

Rewrite the expression.

$\frac{1 (v + 1)}{v + 1} = \frac{A (v (v + 1))}{v} + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.4

Cancel the common factor of $v + 1$ .

Tap for more steps...

Step 6.2.1.1.4.1

Cancel the common factor.

$\frac{1 (v + 1)}{v + 1} = \frac{A (v (v + 1))}{v} + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.4.2

Rewrite the expression.

$1 = \frac{A (v (v + 1))}{v} + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.5

Simplify each term.

Tap for more steps...

Step 6.2.1.1.5.1

Cancel the common factor of $v$ .

Tap for more steps...

Step 6.2.1.1.5.1.1

Cancel the common factor.

$1 = \frac{A (v (v + 1))}{v} + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.5.1.2

Divide $A (v + 1)$ by $1$ .

$1 = A (v + 1) + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.5.2

Apply the distributive property.

$1 = A (v) + A \cdot 1 + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.5.3

Multiply $A$ by $1$ .

$1 = A (v) + A + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.5.4

Cancel the common factor of $v + 1$ .

Tap for more steps...

Step 6.2.1.1.5.4.1

Cancel the common factor.

$1 = A (v) + A + \frac{B (v (v + 1))}{v + 1}$

Step 6.2.1.1.5.4.2

Divide $B v$ by $1$ .

$1 = A v + A + B v$

Step 6.2.1.1.6

Move $A$ .

$1 = A v + B v + A$

Step 6.2.1.2

Create equations for the partial fraction variables and use them to set up a system of equations.

Tap for more steps...

Step 6.2.1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $v$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A + B$

Step 6.2.1.2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $v$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = A$

Step 6.2.1.2.3

Set up the system of equations to find the coefficients of the partial fractions.

$0 = A + B$

$1 = A$

$0 = A + B$

$1 = A$

Step 6.2.1.3

Solve the system of equations.

Tap for more steps...

Step 6.2.1.3.1

Rewrite the equation as $A = 1$ .

$A = 1$

$0 = A + B$

Step 6.2.1.3.2

Replace all occurrences of $A$ with $1$ in each equation.

Tap for more steps...

Step 6.2.1.3.2.1

Replace all occurrences of $A$ in $0 = A + B$ with $1$ .

$0 = (1) + B$

$A = 1$

Step 6.2.1.3.2.2

Simplify the right side.

Tap for more steps...

Step 6.2.1.3.2.2.1

Remove parentheses.

$0 = 1 + B$

$A = 1$

$0 = 1 + B$

$A = 1$

$0 = 1 + B$

$A = 1$

Step 6.2.1.3.3

Solve for $B$ in $0 = 1 + B$ .

Tap for more steps...

Step 6.2.1.3.3.1

Rewrite the equation as $1 + B = 0$ .

$1 + B = 0$

$A = 1$

Step 6.2.1.3.3.2

Subtract $1$ from both sides of the equation.

$B = - 1$

$A = 1$

$B = - 1$

$A = 1$

Step 6.2.1.3.4

Solve the system of equations.

$B = - 1 A = 1$

Step 6.2.1.3.5

List all of the solutions.

$B = - 1, A = 1$

Step 6.2.1.4

Replace each of the partial fraction coefficients in $\frac{A}{v} + \frac{B}{v + 1}$ with the values found for $A$ and $B$ .

$\frac{1}{v} + \frac{- 1}{v + 1}$

Step 6.2.1.5

Move the negative in front of the fraction.

$\int \frac{1}{v} - \frac{1}{v + 1} d v = \int \cos (x) d x$

Step 6.2.2

Split the single integral into multiple integrals.

$\int \frac{1}{v} d v + \int - \frac{1}{v + 1} d v = \int \cos (x) d x$

Step 6.2.3

The integral of $\frac{1}{v}$ with respect to $v$ is $\ln (| v |)$ .

$\ln (| v |) + C_{1} + \int - \frac{1}{v + 1} d v = \int \cos (x) d x$

Step 6.2.4

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

$\ln (| v |) + C_{1} - \int \frac{1}{v + 1} d v = \int \cos (x) d x$

Step 6.2.5

Let $u_{2} = v + 1$ . Then $d u_{2} = d v$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 6.2.5.1

Let $u_{2} = v + 1$ . Find $\frac{d u_{2}}{d v}$ .

Tap for more steps...

Step 6.2.5.1.1

Differentiate $v + 1$ .

$\frac{d}{d v} [v + 1]$

Step 6.2.5.1.2

By the Sum Rule, the derivative of $v + 1$ with respect to $v$ is $\frac{d}{d v} [v] + \frac{d}{d v} [1]$ .

$\frac{d}{d v} [v] + \frac{d}{d v} [1]$

Step 6.2.5.1.3

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

$1 + \frac{d}{d v} [1]$

Step 6.2.5.1.4

Since $1$ is constant with respect to $v$ , the derivative of $1$ with respect to $v$ is $0$ .

$1 + 0$

Step 6.2.5.1.5

Add $1$ and $0$ .

$1$

Step 6.2.5.2

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

$\ln (| v |) + C_{1} - \int \frac{1}{u_{2}} d u_{2} = \int \cos (x) d x$

Step 6.2.6

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\ln (| v |) + C_{1} - (\ln (| u_{2} |) + C_{2}) = \int \cos (x) d x$

Step 6.2.7

Simplify.

$\ln (| v |) - \ln (| u_{2} |) + C_{3} = \int \cos (x) d x$

Step 6.3

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

$\ln (| v |) - \ln (| u_{2} |) + C_{3} = \sin (x) + C_{4}$

Step 6.4

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

$\ln (| v |) - \ln (| u_{2} |) = \sin (x) + C$

Step 7

Solve for $v$ .

Tap for more steps...

Step 7.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| v |}{| u_{2} |}) = \sin (x) + C$

Step 7.2

Reorder $\sin (x)$ and $C$ .

$\ln (\frac{| v |}{| u_{2} |}) = C + \sin (x)$

Step 7.3

To solve for $v$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{| v |}{| u_{2} |})} = e^{C + \sin (x)}$

Step 7.4

Rewrite $\ln (\frac{| v |}{| u_{2} |}) = C + \sin (x)$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{C + \sin (x)} = \frac{| v |}{| u_{2} |}$

Step 7.5

Solve for $v$ .

Tap for more steps...

Step 7.5.1

Rewrite the equation as $\frac{| v |}{| u_{2} |} = e^{C + \sin (x)}$ .

$\frac{| v |}{| u_{2} |} = e^{C + \sin (x)}$

Step 7.5.2

Multiply both sides by $| u_{2} |$ .

$\frac{| v |}{| u_{2} |} | u_{2} | = e^{C + \sin (x)} | u_{2} |$

Step 7.5.3

Simplify the left side.

Tap for more steps...

Step 7.5.3.1

Cancel the common factor of $| u_{2} |$ .

Tap for more steps...

Step 7.5.3.1.1

Cancel the common factor.

$\frac{| v |}{| u_{2} |} | u_{2} | = e^{C + \sin (x)} | u_{2} |$

Step 7.5.3.1.2

Rewrite the expression.

$| v | = e^{C + \sin (x)} | u_{2} |$

Step 7.5.4

Solve for $v$ .

Tap for more steps...

Step 7.5.4.1

Reorder factors in $e^{C + \sin (x)} | u_{2} |$ .

$| v | = | u_{2} | e^{C + \sin (x)}$

Step 7.5.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$v = \pm | u_{2} | e^{C + \sin (x)}$

Step 8

Group the constant terms together.

Tap for more steps...

Step 8.1

Reorder terms.

$v = \pm | u_{2} | e^{\sin (x) + C}$

Step 8.2

Rewrite $e^{\sin (x) + C}$ as $e^{\sin (x)} e^{C}$ .

$v = \pm | u_{2} | (e^{\sin (x)} e^{C})$

Step 8.3

Reorder $e^{\sin (x)}$ and $e^{C}$ .

$v = \pm | u_{2} | (e^{C} e^{\sin (x)})$

Step 8.4

Combine constants with the plus or minus.

$v = C | u_{2} | e^{\sin (x)}$

Step 9

Replace all occurrences of $v$ with $e^{y + \sin (x)}$ .

$e^{y + \sin (x)} = C | u_{2} | e^{\sin (x)}$

Step 10

Solve for $y$ .

Tap for more steps...

Step 10.1

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

$\ln (e^{y + \sin (x)}) = \ln (C | u_{2} | e^{\sin (x)})$

Step 10.2

Expand the left side.

Tap for more steps...

Step 10.2.1

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

$(y + \sin (x)) \ln (e) = \ln (C | u_{2} | e^{\sin (x)})$

Step 10.2.2

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

$(y + \sin (x)) \cdot 1 = \ln (C | u_{2} | e^{\sin (x)})$

Step 10.2.3

Multiply $y + \sin (x)$ by $1$ .

$y + \sin (x) = \ln (C | u_{2} | e^{\sin (x)})$

Step 10.3

Expand the right side.

Tap for more steps...

Step 10.3.1

Rewrite $\ln (C | u_{2} | e^{\sin (x)})$ as $\ln (C | u_{2} |) + \ln (e^{\sin (x)})$ .

$y + \sin (x) = \ln (C | u_{2} |) + \ln (e^{\sin (x)})$

Step 10.3.2

Rewrite $\ln (C | u_{2} |)$ as $\ln (C) + \ln (| u_{2} |)$ .

$y + \sin (x) = \ln (C) + \ln (| u_{2} |) + \ln (e^{\sin (x)})$

Step 10.3.3

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

$y + \sin (x) = \ln (C) + \ln (| u_{2} |) + \sin (x) \ln (e)$

Step 10.3.4

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

$y + \sin (x) = \ln (C) + \ln (| u_{2} |) + \sin (x) \cdot 1$

Step 10.3.5

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

$y + \sin (x) = \ln (C) + \ln (| u_{2} |) + \sin (x)$

Step 10.4

Simplify the right side.

Tap for more steps...

Step 10.4.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$y + \sin (x) = \ln (C | u_{2} |) + \sin (x)$

Step 10.5

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 10.5.1

Subtract $\sin (x)$ from both sides of the equation.

$y = \ln (C | u_{2} |) + \sin (x) - \sin (x)$

Step 10.5.2

Combine the opposite terms in $\ln (C | u_{2} |) + \sin (x) - \sin (x)$ .

Tap for more steps...

Step 10.5.2.1

Subtract $\sin (x)$ from $\sin (x)$ .

$y = \ln (C | u_{2} |) + 0$

Step 10.5.2.2

Add $\ln (C | u_{2} |)$ and $0$ .

$y = \ln (C | u_{2} |)$

Step 11

Use the initial condition to find the value of $C$ by substituting $0$ for $x$ and $0$ for $y$ in $y = \ln (C | u_{2} |)$ .

$0 = \ln (C | u_{2} |)$

Step 12

Solve for $C$ .

Tap for more steps...

Step 12.1

Rewrite the equation as $\ln (C | u_{2} |) = 0$ .

$\ln (C | u_{2} |) = 0$

Step 12.2

To solve for $C$ , rewrite the equation using properties of logarithms.

$e^{\ln (C | u_{2} |)} = e^{0}$

Step 12.3

Rewrite $\ln (C | u_{2} |) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = C | u_{2} |$

Step 12.4

Solve for $C$ .

Tap for more steps...

Step 12.4.1

Rewrite the equation as $C | u_{2} | = e^{0}$ .

$C | u_{2} | = e^{0}$

Step 12.4.2

Anything raised to $0$ is $1$ .

$C | u_{2} | = 1$

Step 12.4.3

Divide each term in $C | u_{2} | = 1$ by $| u_{2} |$ and simplify.

Tap for more steps...

Step 12.4.3.1

Divide each term in $C | u_{2} | = 1$ by $| u_{2} |$ .

$\frac{C | u_{2} |}{| u_{2} |} = \frac{1}{| u_{2} |}$

Step 12.4.3.2

Simplify the left side.

Tap for more steps...

Step 12.4.3.2.1

Cancel the common factor of $| u_{2} |$ .

Tap for more steps...

Step 12.4.3.2.1.1

Cancel the common factor.

$\frac{C | u_{2} |}{| u_{2} |} = \frac{1}{| u_{2} |}$

Step 12.4.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{1}{| u_{2} |}$

Step 13

Substitute $\frac{1}{| u_{2} |}$ for $C$ in $y = \ln (C | u_{2} |)$ and simplify.

Tap for more steps...

Step 13.1

Substitute $\frac{1}{| u_{2} |}$ for $C$ .

$y = \ln (\frac{1}{| u_{2} |} | u_{2} |)$

Step 13.2

Cancel the common factor of $| u_{2} |$ .

Tap for more steps...

Step 13.2.1

Cancel the common factor.

$y = \ln (\frac{1}{| u_{2} |} | u_{2} |)$

Step 13.2.2

Rewrite the expression.

$y = \ln (1)$

Step 13.3

The natural logarithm of $1$ is $0$ .

$y = 0$