What to do
For an initial-value antiderivative problem, you are given a derivative and one condition such as a value of the function at a point. The goal is to find the original function and make it satisfy that condition.
Method
- Integrate the derivative.
Find an antiderivative of the given expression. Include the constant of integration if needed.
- Use the initial value.
Substitute the given point into your antiderivative. This lets you solve for the constant.
- Write the final function.
Replace the constant and simplify the expression completely.
Good habits
- Watch signs carefully when integrating.
- If the derivative contains several terms, integrate term by term.
- Keep the constant until you have used the condition.
Check your work
Differentiate your final answer. It should match the original derivative. Then verify that the function value at the given point matches the initial condition exactly.