Calculus Circus

AP Calculus practice

Integration Practice for AP Calculus

Learn and practice integrals the way AP Calculus teaches them: from reversing derivative rules through definite integrals, accumulation, and differential equations. Calculus Circus has 21 integration skills across 8 topics. 12 skills are free. Start a free skill now, then sign in when you want Calculus Circus to remember your progress.

Start practicing free Sign in to save progress

Sign in to keep completions, track your progress, and make your learning visible to your teacher. Single Term Integration, Polynomial Integration, and Trigonometric Integration are free. Natural log, definite integrals, accumulation, properties of definite integrals, and differential equations require full access — a flat annual fee with no automatic renewal.

What you will practice

An integral undoes a derivative and measures accumulated change. AP Calculus asks students to reverse familiar derivative rules, evaluate definite integrals, read accumulation from graphs, and solve separable differential equations.

∫ x² dx = x³/3 + C. The reverse power rule raises the exponent by one and divides by the new exponent.

The eight integration topics

  1. Single Term Integration

    6 free skills

    Reverse the power rule for one-term functions, including coefficients, negative exponents, fractional exponents, and a mixed set.

    Practice single-term integration

  2. Polynomial Integration

    3 free skills

    Binomials, polynomials, and binomials or trinomials raised to a power.

    Practice polynomial integration

  3. Trigonometric Integration

    3 free skills

    Integrals of sine, cosine, and related trigonometric forms, including chain-rule versions.

    Practice trigonometric integration

  4. Natural Log/Exponential Integration

    3 skills · full access

    Integrals of natural exponential and natural log functions, including binomial log forms.

    See natural log and exponential integration

  5. Definite Integrals

    3 skills · full access

    Evaluate definite integrals of single-term and trigonometric functions.

    See definite integrals

  6. Functions Defined as Integrals

    1 skill · full access

    Read accumulation from a function defined as an integral.

    See functions defined as integrals

  7. Properties of Definite Integrals

    1 skill · full access

    Use additivity and other properties to evaluate definite integrals from functions and graphs.

    See properties of definite integrals

  8. Differential Equations

    1 skill · full access

    Solve separable differential equations.

    See differential equations

How practice works

  • Short problem sets with instant feedback.
  • Skills arranged in a pathway.
  • Single Term Integration, Polynomial Integration, and Trigonometric Integration are free. Sign in to save your progress or join a class created by your teacher.
  • Natural log and exponential integration, definite integrals, functions defined as integrals, properties of definite integrals, and differential equations require full access — a flat annual fee with no automatic renewal. Teachers can see how Calculus Circus works on the About page.

Integration practice FAQ

What integration skills can I practice here?

There are 21 skills across the eight topics above: single-term, polynomial, and trigonometric integration, natural log and exponential integration, definite integrals, accumulation, properties of definite integrals, and differential equations.

Which integration skills are free?

Single Term Integration (6), Polynomial Integration (3), and Trigonometric Integration (3) are free. Natural log and exponential integration (3), definite integrals (3), functions defined as integrals (1), properties of definite integrals (1), and differential equations (1) require full access.

Do I need an account to start?

No. You can begin the free integration skills without an account. Sign in when you want Calculus Circus to keep your completions.

Why should I sign in?

Signing in saves progress so you can return to the next skill and, if you join a class, so a teacher can see where you are practicing.

What comes after integration?

Integration is the last of the three core AP Calculus computational units after limits and derivatives. Join Calculus Circus to get full access to review problems and review flash cards.

Ready to try an integration skill?

Open the integration pathway and start with single-term integrals, or jump to the topic you need. Sign in if you want Calculus Circus to save your place.

Start practicing free Sign in to save progress