AP Calculus AB: Integrals Practice Questions
Test yourself on Integrals with 10 original AP Exams practice questions. Pick an answer to see instant feedback and a full explanation.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of AP Exams.
advertisement
Answer the questions below — you get instant feedback and a full explanation for each.
1. Evaluate the indefinite integral: integral of 6x^2 dx.
Explanation. Use the reverse power rule: integral of x^2 dx is x^3/3, so integral of 6x^2 dx is 6(x^3/3)=2x^3+C.
2. What is the value of integral from 0 to 3 of 4x dx?
Explanation. An antiderivative of 4x is 2x^2. Evaluate from 0 to 3: 2(3^2)-2(0^2)=18.
3. If F'(x)=cos x and F(0)=5, which could be F(x)?
Explanation. An antiderivative of cos x is sin x. Since sin 0=0, F(0)=C=5, so F(x)=sin x+5.
4. Evaluate integral from 1 to 4 of 1/sqrt(x) dx.
Explanation. Rewrite 1/sqrt(x) as x^(-1/2). An antiderivative is 2sqrt(x). From 1 to 4 this gives 2sqrt(4)-2sqrt(1)=4-2=2.
5. A velocity function is v(t)=3t^2-2t feet per second. What is the displacement from t=0 to t=2?
Explanation. Displacement is the definite integral of velocity. An antiderivative is t^3-t^2, and (2^3-2^2)-(0)=8-4=4 feet.
6. Which expression represents the average value of f(x)=x^2 on the interval [0, 3]?
Explanation. The average value of f on [a,b] is (1/(b-a)) times the integral from a to b of f(x) dx. Here b-a=3.
7. Evaluate integral from 0 to pi of sin x dx.
Explanation. An antiderivative of sin x is -cos x. Evaluate: [-cos x] from 0 to pi = -cos(pi)-(-cos 0)=1-(-1)=2.
8. If integral from 2 to 5 of f(x) dx = 7, what is integral from 5 to 2 of f(x) dx?
Explanation. Reversing the bounds of a definite integral changes the sign. Therefore integral from 5 to 2 of f(x) dx = -7.
9. Using the Fundamental Theorem of Calculus, if G(x)=integral from 1 to x of (t^3+2t) dt, what is G'(x)?
Explanation. For G(x)=integral from a to x of f(t) dt, the derivative is G'(x)=f(x). Substitute x into the integrand to get x^3+2x.
10. Evaluate integral of e^(2x) dx.
Explanation. Because d/dx[e^(2x)] = 2e^(2x), the antiderivative of e^(2x) is (1/2)e^(2x)+C.
📘 Want a full structured course and official-style practice tests? Browse top-rated AP Exams prep books and courses. Some links are affiliate links; we may earn a commission at no cost to you.
advertisement
FAQ
What integral rules should I know for AP Calculus AB?
Know the reverse power rule, basic trigonometric and exponential antiderivatives, definite integral properties, average value, and the Fundamental Theorem of Calculus.
How are definite integrals different from indefinite integrals?
An indefinite integral gives a family of antiderivatives plus C. A definite integral gives a number that can represent net area, accumulation, displacement, or total change.
What is a common mistake on AP integral questions?
Students often forget the constant C on indefinite integrals, reverse bounds without changing the sign, or treat total distance and displacement as the same quantity.