# Calculus Examples

,

Step 1

Step 1.1

Divide each term in by and simplify.

Step 1.1.1

Divide each term in by .

Step 1.1.2

Simplify the left side.

Step 1.1.2.1

Cancel the common factor of .

Step 1.1.2.1.1

Cancel the common factor.

Step 1.1.2.1.2

Divide by .

Step 1.1.3

Simplify the right side.

Step 1.1.3.1

Move the negative in front of the fraction.

Step 1.2

Combine the numerators over the common denominator.

Step 1.3

Multiply both sides by .

Step 1.4

Cancel the common factor of .

Step 1.4.1

Cancel the common factor.

Step 1.4.2

Rewrite the expression.

Step 1.5

Rewrite the equation.

Step 2

Step 2.1

Set up an integral on each side.

Step 2.2

Integrate the left side.

Step 2.2.1

Let . Then , so . Rewrite using and .

Step 2.2.1.1

Let . Find .

Step 2.2.1.1.1

Differentiate .

Step 2.2.1.1.2

By the Sum Rule, the derivative of with respect to is .

Step 2.2.1.1.3

Evaluate .

Step 2.2.1.1.3.1

Since is constant with respect to , the derivative of with respect to is .

Step 2.2.1.1.3.2

Differentiate using the Power Rule which states that is where .

Step 2.2.1.1.3.3

Multiply by .

Step 2.2.1.1.4

Differentiate using the Constant Rule.

Step 2.2.1.1.4.1

Since is constant with respect to , the derivative of with respect to is .

Step 2.2.1.1.4.2

Add and .

Step 2.2.1.2

Rewrite the problem using and .

Step 2.2.2

Simplify.

Step 2.2.2.1

Multiply by .

Step 2.2.2.2

Move to the left of .

Step 2.2.3

Since is constant with respect to , move out of the integral.

Step 2.2.4

The integral of with respect to is .

Step 2.2.5

Simplify.

Step 2.2.6

Replace all occurrences of with .

Step 2.3

The integral of with respect to is .

Step 2.4

Group the constant of integration on the right side as .

Step 3

Step 3.1

Multiply both sides of the equation by .

Step 3.2

Simplify both sides of the equation.

Step 3.2.1

Simplify the left side.

Step 3.2.1.1

Simplify .

Step 3.2.1.1.1

Combine and .

Step 3.2.1.1.2

Cancel the common factor of .

Step 3.2.1.1.2.1

Cancel the common factor.

Step 3.2.1.1.2.2

Rewrite the expression.

Step 3.2.2

Simplify the right side.

Step 3.2.2.1

Apply the distributive property.

Step 3.3

Move all the terms containing a logarithm to the left side of the equation.

Step 3.4

Simplify the left side.

Step 3.4.1

Simplify .

Step 3.4.1.1

Simplify each term.

Step 3.4.1.1.1

Simplify by moving inside the logarithm.

Step 3.4.1.1.2

Remove the absolute value in because exponentiations with even powers are always positive.

Step 3.4.1.2

Use the quotient property of logarithms, .

Step 3.5

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

Step 3.6

Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .

Step 3.7

Solve for .

Step 3.7.1

Rewrite the equation as .

Step 3.7.2

Multiply both sides by .

Step 3.7.3

Simplify.

Step 3.7.3.1

Simplify the left side.

Step 3.7.3.1.1

Cancel the common factor of .

Step 3.7.3.1.1.1

Cancel the common factor.

Step 3.7.3.1.1.2

Rewrite the expression.

Step 3.7.3.2

Simplify the right side.

Step 3.7.3.2.1

Reorder factors in .

Step 3.7.4

Solve for .

Step 3.7.4.1

Remove the absolute value term. This creates a on the right side of the equation because .

Step 3.7.4.2

Add to both sides of the equation.

Step 3.7.4.3

Divide each term in by and simplify.

Step 3.7.4.3.1

Divide each term in by .

Step 3.7.4.3.2

Simplify the left side.

Step 3.7.4.3.2.1

Cancel the common factor of .

Step 3.7.4.3.2.1.1

Cancel the common factor.

Step 3.7.4.3.2.1.2

Divide by .

Step 4

Step 4.1

Simplify the constant of integration.

Step 4.2

Combine constants with the plus or minus.

Step 5

Use the initial condition to find the value of by substituting for and for in .

Step 6

Step 6.1

Rewrite the equation as .

Step 6.2

Simplify each term.

Step 6.2.1

One to any power is one.

Step 6.2.2

Multiply by .

Step 6.3

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

Step 6.3.1

Subtract from both sides of the equation.

Step 6.3.2

Write as a fraction with a common denominator.

Step 6.3.3

Combine the numerators over the common denominator.

Step 6.3.4

Subtract from .

Step 6.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

Step 7

Step 7.1

Substitute for .

Step 7.2

Multiply by .