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Indefinite Integration

An antiderivative matched to a printed form or evaluated at a point, so the constant matters. Choosing the substitution is the whole question.

Questions in the bank
46
q/paper in 2025–26
0.41
Numeric answer
30%
Notes pages
4

Tier: Long tail

When you’ll see it

An antiderivative is asked for, usually with a given value to fix the constant or a printed form whose coefficients must be read off.

How this chapter is tested

Questions rarely stop at the antiderivative. A given value fixes the constant and the question asks for the value at another point, or the result is matched to a printed form such as A ln|…| + B tan⁻¹(…) + C and the question asks for A + B. So the integral must be exact, constant and all.

The first three pages reduce an integrand to a standard form: a substitution that clears a root or a high power of x, partial fractions or a completed square, and trigonometric integrands turned into algebra with t = tan x or t = sin x ± cos x. The last page is integration by parts, the pattern eˣ(f + f′), and integrands that are already the derivative of a product or a quotient.

Choosing t is the whole question; once it is right, the integral is a power of t. When stuck, differentiating each option is often quicker than integrating, and it checks the answer either way. The same techniques run through Definite Integration and Differential Equations.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Algebraic substitution

    Put t equal to a root to clear it; for two linear factors use their ratio; take a power of x out of a long expression.

  • Rational functions and standard forms

    Divide first if the top's degree is not lower, then split into partial fractions, complete the square, or put t = x ± 1/x.

  • Trigonometric integrals

    Divide through to write everything in tan x or cot x, or put t = sin x, cos x, or sin x ± cos x.

  • Integration by parts and reverse differentiation

    ∫u dv = uv − ∫v du; ∫eˣ(f + f′) dx = eˣ f + C; spot a product or quotient derivative in the integrand.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Given values are in x

    With x = t⁶, the point x = 64 is t = 2, not t = 64. Convert the point before fixing the constant.

  • Match the substitution to the top

    For x² + 1 on top put t = x − 1/x; for x² − 1 put t = x + 1/x. The other choice leaves no dt.

  • Signs alternate in repeated parts

    ∫x² cos x dx = x² sin x + 2x cos x − 2 sin x + C. One wrong sign changes every value computed from it.

  • Cot brings a minus sign

    With t = cot x, dt = −cosec²x dx, so every term of the answer changes sign.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Indefinite Integration notes

Drill every Indefinite Integration question

46 questions from the bank, across 4 subtopics.

Drill the 46 questions

Drill one subtopic at a time

The 4 subtopics, in teaching order.

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