How do I choose u in integration by parts?

Choose u to be the factor that becomes simpler when differentiated. A common order of preference is logarithm, algebraic (powers of x), trigonometric, exponential: so in ∫ x e²ˣ dx take u = x, and in ∫ x ln x dx take u = ln x.

How many times can integration by parts be used in DSE M2?

At most twice in finding one integral. That is a stated limit in the curriculum notes, so an integral needing three applications is outside the syllabus.

Do I need to integrate inverse trigonometric functions?

No. You need the notations sin⁻¹x, cos⁻¹x and tan⁻¹x because they appear in answers from trigonometric substitution, but integrands containing them are not required.

Indefinite Integration (DSE M2): Substitution, Integration by Parts and Trigonometric Substitution

Indefinite integration reverses differentiation: ∫ f(x) dx = F(x) + C where F′(x) = f(x). DSE M2 needs three techniques beyond the basic formulas: substitution, integration by parts ∫ u dv = uv − ∫ v du (used at most twice), and trigonometric substitution.

Solve Indefinite Integration questions with AI

The family y = x² − x + C from Example 1. Each value of C gives a different curve with the same slope 2x − 1; only C = 3 passes through (1, 3). Try it yourself →

Integration Formulas (DSE M2)

Power∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1)

Reciprocal∫ (1/x) dx = ln|x| + C

Exponential∫ eˣ dx = eˣ + C

Trigonometric∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C, ∫ sec² x dx = tan x + C

Integration by parts∫ u dv = uv − ∫ v du

√(a² − x²) or 1/√(a² − x²)Let x = a sin θ

1/(x² + a²)Let x = a tan θ

Worked Examples — Indefinite Integration

Example 1: Finding a curve from its slope

The slope of a curve at any point (x, y) is 2x − 1, and the curve passes through (1, 3). Find the equation of the curve.

dy/dx = 2x − 1
y = ∫ (2x − 1) dx = x² − x + C
Substitute (1, 3): 3 = 1 − 1 + C, so C = 3

Answer: y = x² − x + 3

Example 2: Integration by parts

Find ∫ x e²ˣ dx.

Let u = x, dv = e²ˣ dx, so du = dx, v = ½e²ˣ
∫ x e²ˣ dx = ½x e²ˣ − ∫ ½e²ˣ dx
= ½x e²ˣ − ¼e²ˣ + C

Answer: ½x e²ˣ − ¼e²ˣ + C

Example 3: Trigonometric substitution

Find ∫ 1/(x² + 4) dx.

Let x = 2 tan θ, so dx = 2 sec² θ dθ and x² + 4 = 4 sec² θ
∫ (2 sec² θ) / (4 sec² θ) dθ = ∫ ½ dθ = ½θ + C
θ = tan⁻¹(x/2)

Answer: ½ tan⁻¹(x/2) + C

💡 Exam Traps & Secrets

⚠️ Forgetting + CEvery indefinite integral needs the constant of integration. It is the single most common lost mark in M2 calculus, and it matters: the constant is what you solve for when the curve passes through a given point.

⚠️ Substitution: change dx as wellWith u = g(x) you must replace dx using du = g′(x) dx, and the final answer must be back in terms of x. Leaving a mixture of x and u inside one integral is not valid working.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Integration by parts)Find the integral of x cos x dx.

(Substitution)Find the integral of x √(x² + 1) dx.

(Trigonometric substitution)Find the integral of 1/√(9 − x²) dx using a trigonometric substitution.