# Calculus Examples

,

Divide each term in by and simplify.

Combine the numerators over the common denominator.

Multiply both sides by .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Rewrite the equation.

Set up an integral on each side.

Integrate the left side.

Let . Then , so . Rewrite using and .

Let . Find .

Differentiate .

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

Evaluate .

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

Differentiate using the Power Rule which states that is where .

Multiply by .

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

Add and .

Rewrite the problem using and .

Simplify.

Multiply by .

Move to the left of .

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

The integral of with respect to is .

Simplify.

Replace all occurrences of with .

The integral of with respect to is .

Group the constant of integration on the right side as .

Multiply both sides of the equation by .

Simplify both sides of the equation.

Simplify the left side.

Simplify .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Simplify the right side.

Apply the distributive property.

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

Simplify the left side.

Simplify .

Simplify each term.

Simplify by moving inside the logarithm.

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

Use the quotient property of logarithms, .

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

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

Solve for .

Rewrite the equation as .

Multiply both sides by .

Simplify.

Simplify the left side.

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Simplify the right side.

Reorder factors in .

Solve for .

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

Add to both sides of the equation.

Divide each term in by and simplify.

Divide each term in by .

Simplify the left side.

Cancel the common factor of .

Cancel the common factor.

Divide by .

Simplify the constant of integration.

Combine constants with the plus or minus.

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

Rewrite the equation as .

Simplify each term.

One to any power is one.

Multiply by .

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

Subtract from both sides of the equation.

Write as a fraction with a common denominator.

Combine the numerators over the common denominator.

Subtract from .

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

Substitute.

Multiply by .