Integration techniques: substitution, parts, partial fractions

IB Mathematics AA HL· 5.9 Core integration techniques· 20 min read

1. U-Substitution for Composite Integrands★★☆☆☆⏱ 6 min

📘 Definition

U-Substitution

$u = g (x)$

A reverse chain rule technique for integrating composite functions of the form $f (g (x)) g^{'} (x)$ . It simplifies the integral by substituting $u$ for the inner function $g (x)$ .

Example:

For $e x t cos (x^{2}) im es 2 x$ , $u = x^{2}$

📐 Worked Example

Find the indefinite integral $ \int 2x \cos(x^2) dx $,

1
1. Identify the inner function, whose derivative appears as a factor in the integrand: $g (x) = x^{2}$ , $g^{'} (x) = 2 x$ which matches the extra factor.
2
1. Substitute $u = x^{2}$ , which gives:
3
$\frac{du}{dx} = 2x \implies du = 2x dx$
4
1. Rewrite the integral in terms of $u$ :
5
$\int \cos(u) du$
6
1. Integrate with respect to $u$ :
7
$\sin(u) + C$
8
1. Substitute back $u = x^{2}$ to get the final result:
9
$\sin(x^2) + C$

2. Integration by Parts for Product Integrals★★★☆☆⏱ 7 min

📘 Definition

Integration by Parts

$\int u d v = uv - \int v d u$

A technique derived from the product rule for differentiation, used to integrate products of functions that cannot be simplified via substitution.

📐 Worked Example

Evaluate $ \int x e^x dx $,

1
1. Choose $u$ as the function that simplifies when differentiated, and $d v$ as the function that is easy to integrate. We let:
2
$u = x, \quad dv = e^x dx$
3
1. Calculate $d u$ (differentiate $u$ ) and $v$ (integrate $d v$ ):
4
$du = dx, \quad v = e^x$
5
1. Apply the integration by parts formula:
6
$uv - \int v \ du = x e^x - \int e^x dx$
7
1. Integrate the remaining simpler integral and simplify:
8
$x e^x - e^x + C = e^x(x - 1) + C$

Exam tip:

Always show your substitution of $u, d v, d u, v$ in exams to earn full method marks even if you make a small arithmetic error.

3. Partial Fractions for Rational Integrands★★★★☆⏱ 7 min

📘 Definition

Partial Fraction Decomposition

A method that splits a proper rational function (degree of numerator < degree of denominator) into a sum of simpler fractions that can be integrated term-by-term.

📐 Worked Example

Find $ \int \frac{3x + 1}{x^2 + x - 2} dx $,

1
1. Factorize the denominator:
2
$x^2 + x - 2 = (x + 2)(x - 1)$
3
1. Set up the partial fraction decomposition for distinct linear factors:
4
$\frac{3x + 1}{(x+2)(x-1)} = \frac{A}{x+2} + \frac{B}{x-1}$
5
1. Multiply both sides by the denominator, then solve for constants $A$ and $B$ :
6
$3x + 1 = A(x-1) + B(x+2)$
7
Substitute $x = 1$ : $4 = 3 B ⟹ B = \frac{4}{3}$ . Substitute $x = - 2$ : $- 5 = - 3 A ⟹ A = \frac{5}{3}$ .
8
1. Rewrite the integral and integrate term-by-term:
9
$\int \left( \frac{5}{3(x+2)} + \frac{4}{3(x-1)} \right) dx$
10
$\frac{5}{3} \ln|x+2| + \frac{4}{3} \ln|x-1| + C$

4. Common Pitfalls

Wrong move:

Forgetting to change limits for u-substitution on definite integrals

Why:

You will evaluate the u-antiderivative at the original x limits, which gives an incorrect result

Correct move:

Either change limits to match u, or substitute back to x before evaluating the definite integral

Wrong move:

Choosing the wrong u in integration by parts

Why:

The resulting integral becomes more complicated than the original and cannot be solved

Correct move:

Follow the LIATE mnemonic to select the correct u for the problem

Wrong move:

Skipping polynomial division when numerator degree ≥ denominator degree for partial fractions

Why:

The partial fraction decomposition will be set up incorrectly, leading to wrong coefficients

Correct move:

Always compare degrees first, divide to get a polynomial plus a proper rational function before decomposing

Wrong move:

Forgetting the constant of integration $C$ for indefinite integrals

Why:

Loses easy method marks in the exam and leaves the antiderivative incomplete

Correct move:

Always add $+ C$ at the end of any indefinite integral calculation

Wrong move:

Omitting absolute value signs when integrating $\frac{1}{u}$

Why:

The natural logarithm is only defined for positive arguments, so the solution is incomplete

Correct move:

Always write the result as $ln ∣ u ∣ + C$ when integrating $\frac{1}{u}$

5. Quick Reference Cheatsheet

Technique	When to use	Core formula/rule
U-Substitution	Composite $f (g (x)) g^{'} (x)$	$\int f (g (x)) g^{'} (x) d x = \int f (u) d u, u = g (x)$
Integration by Parts	Product of two functions	$\int u d v = uv - \int v d u$
Partial Fractions	Proper rational functions	Decompose to simple fractions, integrate term-by-term
LIATE Rule (for by parts)	Choosing $u$	Log > Inv Trig > Algebra > Trig > Exp

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · P1
Evaluate definite integral by parts
2022 · P2
Combine three integration techniques
2021 · P1
Integrate using partial fractions

Going deeper

What's Next

These three integration techniques form the foundation for all advanced integration work in IB AA HL. You will use them constantly in applications like finding areas between curves, calculating volumes of revolution, and solving first and second order differential equations. Mastering the ability to select the correct technique for a given integral is one of the most important skills for exam success, as most non-trivial integral questions will require you to identify the right approach independently, and many will require combining multiple methods.