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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
Split the single integral into multiple integrals.
Step 2.3.2
Since is constant with respect to , move out of the integral.
Step 2.3.3
By the Power Rule, the integral of with respect to is .
Step 2.3.4
Since is constant with respect to , move out of the integral.
Step 2.3.5
By the Power Rule, the integral of with respect to is .
Step 2.3.6
Apply the constant rule.
Step 2.3.7
Simplify.
Step 2.3.7.1
Simplify.
Step 2.3.7.1.1
Combine and .
Step 2.3.7.1.2
Combine and .
Step 2.3.7.2
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
One to any power is one.
Step 4.2.1.2
One to any power is one.
Step 4.2.1.3
Multiply by .
Step 4.2.1.4
Multiply by .
Step 4.2.2
Simplify by subtracting numbers.
Step 4.2.2.1
Subtract from .
Step 4.2.2.2
Add and .
Step 4.3
Move all terms not containing to the right side of the equation.
Step 4.3.1
Subtract from both sides of the equation.
Step 4.3.2
Subtract from .
Step 5
Step 5.1
Substitute for .