Integration Techniques You Must Know for AP Calculus BC | RevisionDojo

In AP Calculus BC, integration is just as important as differentiation — and often more challenging. While AP Calculus AB focuses mainly on basic antiderivatives and the Fundamental Theorem of Calculus, BC pushes further into advanced integration techniques that require strategic thinking and multiple steps.

In this RevisionDojo guide, you’ll learn:

• The must-know BC integration techniques
• When and how to use each method
• Step-by-step examples
• Common pitfalls to avoid

📚 1. u-Substitution (Reverse Chain Rule)

When to Use:

• Integrand contains a function and its derivative (or something close to it)

Formula:
If u=g(x)u = g(x), then:

∫f(g(x))⋅g′(x) dx=∫f(u) du\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du

Example:

∫2x⋅ex2dx\int 2x \cdot e^{x^2} dx

Let u=x2u = x^2, du=2x dxdu = 2x \, dx →

∫eudu=eu+C=ex2+C\int e^u du = e^u + C = e^{x^2} + C

📚 2. Integration by Parts

When to Use:

• Product of two functions where u-substitution doesn’t work well

Formula:

∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du

Example:

∫xexdx\int x e^x dx

Let u=xu = x, dv=exdxdv = e^x dx → du=dxdu = dx, v=exv = e^x

xex−∫exdx=xex−ex+Cx e^x - \int e^x dx = x e^x - e^x + C

📚 3. Partial Fractions

When to Use:

• Rational functions where degree of numerator < degree of denominator

Example:

1x2−1=Ax−1+Bx+1\frac{1}{x^2 - 1} = \frac{A}{x - 1} + \frac{B}{x + 1}

Integrate each term separately after finding A and B.

📚 4. Trigonometric Integrals

When to Use:

• Powers of sin and cos, or sec and tan

Example:

∫sin⁡3xcos⁡xdx\int \sin^3 x \cos x dx

Let u=sin⁡xu = \sin x, du=cos⁡xdxdu = \cos x dx →

∫u3du=u44+C=sin⁡4x4+C\int u^3 du = \frac{u^4}{4} + C = \frac{\sin^4 x}{4} + C

📚 5. Trigonometric Substitution

When to Use:

• Integrals involving a2−x2\sqrt{a^2 - x^2}, a2+x2\sqrt{a^2 + x^2}, or x2−a2\sqrt{x^2 - a^2}

Example:
If a2−x2\sqrt{a^2 - x^2} appears, substitute x=asin⁡θx = a \sin \theta.

📚 6. Improper Integrals

When to Use:

• Infinite limits or discontinuities in the interval

Example:

∫1∞1x2dx=lim⁡b→∞[−1x]1b\int_1^\infty \frac{1}{x^2} dx = \lim_{b \to \infty} \left[ -\frac{1}{x} \right]_1^b

⚠️ Common Mistakes to Avoid

• Forgetting to revert back to xx after u-substitution
• Skipping absolute value in ln⁡∣u∣\ln|u| results
• Mixing up integration by parts and product rule (different processes!)
• Not checking convergence for improper integrals

📊 Practice Strategy from RevisionDojo

• Master each technique separately before mixing them
• Use past AP Calculus BC FRQs that combine multiple methods in one problem
• Drill speed by timing yourself on common integrals

🧭 Final Advice from RevisionDojo

Integration in AP Calculus BC isn’t about memorizing every possible integral — it’s about recognizing patterns and choosing the right tool quickly.
With consistent practice, you’ll turn even intimidating integrals into step-by-step solutions.