# Calculus Examples

,

Step 1

Rewrite the equation.

Step 2

Step 2.1

Set up an integral on each side.

Step 2.2

Apply the constant rule.

Step 2.3

Integrate the right side.

Step 2.3.1

Let . Then , so . Rewrite using and .

Step 2.3.1.1

Let . Find .

Step 2.3.1.1.1

Differentiate .

Step 2.3.1.1.2

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

Step 2.3.1.1.3

Differentiate using the Power Rule which states that is where .

Step 2.3.1.1.4

Multiply by .

Step 2.3.1.2

Rewrite the problem using and .

Step 2.3.2

Simplify.

Step 2.3.2.1

Move the negative in front of the fraction.

Step 2.3.2.2

Combine and .

Step 2.3.3

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

Step 2.3.4

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

Step 2.3.5

The integral of with respect to is .

Step 2.3.6

Simplify.

Step 2.3.7

Replace all occurrences of with .

Step 2.4

Group the constant of integration on the right side as .

Step 3

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

Step 4

Step 4.1

Rewrite the equation as .

Step 4.2

Simplify each term.

Step 4.2.1

Multiply by .

Step 4.2.2

Anything raised to is .

Step 4.2.3

Multiply by .

Step 4.3

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

Step 4.3.1

Add to both sides of the equation.

Step 4.3.2

To write as a fraction with a common denominator, multiply by .

Step 4.3.3

Combine and .

Step 4.3.4

Combine the numerators over the common denominator.

Step 4.3.5

Simplify the numerator.

Step 4.3.5.1

Multiply by .

Step 4.3.5.2

Add and .

Step 5

Step 5.1

Substitute for .

Step 5.2

Combine and .