Calculus Examples
,
Step 1
Step 1.1
Add to both sides of the equation.
Step 1.2
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
Split the single integral into multiple integrals.
Step 2.3.2
By the Power Rule, the integral of with respect to is .
Step 2.3.3
By the Power Rule, the integral of with respect to is .
Step 2.3.4
Simplify.
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 .
Step 4.2.1
Simplify each term.
Step 4.2.1.1
Raise to the power of .
Step 4.2.1.2
Combine and .
Step 4.2.1.3
Move the negative in front of the fraction.
Step 4.2.1.4
Raise to the power of .
Step 4.2.1.5
Cancel the common factor of .
Step 4.2.1.5.1
Factor out of .
Step 4.2.1.5.2
Cancel the common factor.
Step 4.2.1.5.3
Rewrite the expression.
Step 4.2.2
To write as a fraction with a common denominator, multiply by .
Step 4.2.3
Combine and .
Step 4.2.4
Combine the numerators over the common denominator.
Step 4.2.5
Simplify the numerator.
Step 4.2.5.1
Multiply by .
Step 4.2.5.2
Add and .
Step 4.2.6
Move the negative in front of the fraction.
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
Simplify each term.
Step 5.2.1
Combine and .
Step 5.2.2
Combine and .