Integration by Parts

∫ Integration by Parts

Part 1 of 7 — The Formula & LIATE Rule

Integration by parts is the integration counterpart of the product rule for derivatives. It’s essential for AP Calculus BC and appears on virtually every exam.

The Integration by Parts Formula

Starting from the product rule: $\frac{d}{dx}[u \cdot v] = u\frac{dv}{dx} + v\frac{du}{dx}$

Integrating both sides and rearranging:

$\boxed{\int u\,dv = uv - \int v\,du}$

Think of it as “swapping” one integral for a (hopefully) simpler one.

Key Fact: Integration by parts is the go-to technique when the integrand is a product of two different types of functions (e.g., polynomial × exponential).

The LIATE Rule for Choosing $u$

The hardest part is deciding which factor to call $u$ and which to call $dv$. Use the LIATE priority:

AP Tip: LIATE works for ~95% of IBP problems. The idea: $u$ should get simpler when differentiated, while $dv$ should be easy to integrate.

Worked Example — $\int x e^x\,dx$

$\boxed{\int x e^x\,dx = e^x(x - 1) + C}$

Verification: $\frac{d}{dx}[e^x(x-1)] = e^x(x-1) + e^x = xe^x$ ✔

Applying the Formula

LIATE Selection Practice

Compute an IBP Integral

Key Takeaways — Part 1

Coming Up: Part 2 introduces the Tabular Method — a shortcut for polynomial × exponential/trig integrals that eliminates repetitive IBP steps.

∫ Integration by Parts

Part 2 of 7 — Tabular (Column) Method

The tabular method is a shortcut for integrating products of the form (polynomial) × (easy-to-integrate function). Instead of repeatedly applying IBP, you organize everything in a table.

How the Tabular Method Works

1. Place the polynomial in the “Differentiate” column (it eventually reaches 0)
2. Place the other factor in the “Integrate” column
3. Alternate signs: $+, -, +, -, \ldots$
4. Multiply diagonally and sum

Example: $\int x^2 e^x\,dx$

Reading diagonally: $+x^2 e^x - 2xe^x + 2e^x + C$

$\boxed{\int x^2 e^x\,dx = e^x(x^2 - 2x + 2) + C}$

Key Fact: The tabular method works whenever one factor differentiates to zero (polynomials). It saves enormous time on the AP exam.

Tabular Method with Trig

Example: $\int x^3 \cos x\,dx$

$\int x^3\cos x\,dx = x^3\sin x + 3x^2\cos x - 6x\sin x - 6\cos x + C$

When Does Tabular NOT Work?

Tabular Method Practice

Identify the Method

Tabular Computation

Key Takeaways — Part 2

Coming Up: Part 3 covers cycling problems — what happens when IBP brings the original integral back (the boomerang technique).

∫ Integration by Parts

Part 3 of 7 — Cycling (Boomerang) Problems

Some IBP integrals don’t simplify — instead, after two applications the original integral reappears. When this happens, you solve algebraically for the unknown integral.

The Boomerang Technique

When does cycling occur? When both factors regenerate under repeated differentiation and integration:

• $e^{ax} \cdot \sin(bx)$ or $e^{ax} \cdot \cos(bx)$

Full Worked Example: $\int e^x \sin x\,dx$

Let $I = \int e^x \sin x\,dx$

IBP #1: $u = \sin x$, $dv = e^x\,dx$ $I = e^x \sin x - \int e^x \cos x\,dx$

IBP #2: $u = \cos x$, $dv = e^x\,dx$ $I = e^x \sin x - \left[e^x \cos x - \int e^x(-\sin x)\,dx\right]$ $I = e^x \sin x - e^x \cos x - \int e^x \sin x\,dx$ $I = e^x \sin x - e^x \cos x - I$

Solve for $I$: $2I = e^x(\sin x - \cos x)$

$\boxed{\int e^x \sin x\,dx = \frac{e^x(\sin x - \cos x)}{2} + C}$

Key Fact: You MUST use the same type of choice for $u$ in both applications (both times pick trig, or both times pick exponential). Mixing causes an infinite loop.

General Formula for $e^{ax}$ × Trig

$\boxed{\int e^{ax}\sin(bx)\,dx = \frac{e^{ax}(a\sin(bx) - b\cos(bx))}{a^2 + b^2} + C}$

$\boxed{\int e^{ax}\cos(bx)\,dx = \frac{e^{ax}(a\cos(bx) + b\sin(bx))}{a^2 + b^2} + C}$

AP Tip: Memorizing the general formula can save 3–4 minutes on a free-response question. But you should know how to derive it via IBP cycling.

Cycling IBP Practice

Identify the Technique

Cycling Computation

Key Takeaways — Part 3

Coming Up: Part 4 applies IBP to definite integrals — including evaluating bounds correctly.

∫ Integration by Parts

Part 4 of 7 — Definite Integrals with IBP

On the AP exam, many IBP problems involve definite integrals. You can either find the antiderivative first, then evaluate at the bounds, or carry the bounds through the entire process.

Definite Integral IBP Formula

$\boxed{\int_a^b u\,dv = [uv]_a^b - \int_a^b v\,du}$

Strategy Options

AP Tip: On free-response questions, show each step clearly. Write the $[uv]_a^b$ term explicitly before evaluating.

Worked Example 1: $\int_0^1 xe^x\,dx$

$\boxed{\int_0^1 xe^x\,dx = 1}$

Worked Example 2: $\int_1^e (\ln x)^2\,dx$

$\boxed{\int_1^e (\ln x)^2\,dx = e - 2}$

Definite IBP Practice

Step-by-Step Evaluation

Exact Computation

Key Takeaways — Part 4

Key Fact: Many definite IBP integrals yield surprisingly clean answers. Always simplify fully before reporting your answer.

Coming Up: Part 5 covers special cases including inverse trig and logarithmic IBP integrals.

∫ Integration by Parts

Part 5 of 7 — Special Cases: Inverse Trig & Logarithms

Some functions don’t have obvious antiderivatives, but they DO have known derivatives. For these, we set the tricky function as $u$ and let $dv = dx$.

The “$dv = dx$” Strategy

When the integrand has no obvious product structure, set:

• $u =$ the function (so you can differentiate it)
• $dv = dx$ (so $v = x$)

This works beautifully for:

Key Fact: L and I in LIATE always become $u$. Their derivatives produce algebraic expressions that pair nicely with $v = x$.

Essential Results

1. $\int \ln x\,dx$

$u = \ln x$, $dv = dx \Rightarrow du = \frac{1}{x}\,dx$, $v = x$

$\int \ln x\,dx = x\ln x - \int x \cdot \frac{1}{x}\,dx = x\ln x - x + C$

$\boxed{\int \ln x\,dx = x(\ln x - 1) + C}$

2. $\int \arctan x\,dx$

$u = \arctan x$, $dv = dx \Rightarrow du = \frac{1}{1+x^2}\,dx$, $v = x$

$\int \arctan x\,dx = x\arctan x - \int \frac{x}{1+x^2}\,dx$

The remaining integral: let $w = 1+x^2$, $dw = 2x\,dx$:

$\boxed{\int \arctan x\,dx = x\arctan x - \frac{1}{2}\ln(1+x^2) + C}$

3. $\int \arcsin x\,dx$

$u = \arcsin x$, $dv = dx \Rightarrow du = \frac{1}{\sqrt{1-x^2}}\,dx$, $v = x$

$\int \arcsin x\,dx = x\arcsin x - \int \frac{x}{\sqrt{1-x^2}}\,dx$

Let $w = 1 - x^2$: remaining integral $= \sqrt{1-x^2}$

$\boxed{\int \arcsin x\,dx = x\arcsin x + \sqrt{1-x^2} + C}$

Higher Powers of $\ln x$

$\int (\ln x)^2\,dx$

$u = (\ln x)^2$, $dv = dx$

$= x(\ln x)^2 - 2\int \ln x\,dx = x(\ln x)^2 - 2(x\ln x - x) + C$

$\boxed{\int (\ln x)^2\,dx = x[(\ln x)^2 - 2\ln x + 2] + C}$

Reduction formula (for reference): $\int (\ln x)^n\,dx = x(\ln x)^n - n\int (\ln x)^{n-1}\,dx$

Special Cases Practice

Identify the Setup

Numerical Evaluation

Key Takeaways — Part 5

AP Tip: These results are worth memorizing — they appear frequently in free-response questions and save substantial time.

Coming Up: Part 6 is a problem-solving workshop with mixed IBP challenges.

∫ Integration by Parts

Part 6 of 7 — Problem-Solving Workshop

This part is a mixed-practice workshop. Every problem requires identifying the correct IBP approach, then executing it. Think before you compute!

Decision Flowchart

Mixed IBP Practice — Round 1

Mixed IBP Practice — Round 2

Method Identification

Definite Integral Challenge

Key Takeaways — Part 6

AP Tip: If you get stuck, try a different $u$/$dv$ split. There’s often more than one path to the answer.

Coming Up: Part 7 is the comprehensive review and assessment covering all IBP techniques.

∫ Integration by Parts — Review

Part 7 of 7 — Comprehensive Review & Assessment

This final part tests your mastery of all IBP techniques: the basic formula, LIATE, tabular method, cycling, definite integrals, and special cases.

Complete Reference Table

Assessment — Conceptual

Assessment — Computational

Method Selection Review

Final Calculation

Integration by Parts — Complete! ✅

You’ve mastered:

1. ✔ The IBP formula and LIATE rule
2. ✔ Tabular method for polynomial × exp/trig
3. ✔ Cycling technique for $e^{ax}$ × trig
4. ✔ Definite integrals with IBP
5. ✔ Special cases (inverse trig, logs)
6. ✔ Mixed problem identification

AP Exam Frequency

Key Fact: Integration by parts appears on every AP Calculus BC exam. Master all five approaches and you’ll handle any IBP problem confidently.