IB Maths AI HL
Further Integration
Paper 1 & 2
~5 min read
Integrating Special Functions
The power rule can’t touch 1x, ex, sin or cos — these are special functions with their own standard integrals. Learn the short list, reverse the chain rule for an (ax + b) inside, and the rest is the usual integration routine.
📘 What you need to know
- The reciprocal: ∫ 1x dx = ln|x| + c — the power rule fails here (it would divide by 0).
- Exponential: ∫ ex dx = ex + c — it integrates to itself.
- Trig: ∫ sin x dx = −cos x + c and ∫ cos x dx = sin x + c.
- Linear inside: for e(ax+b), sin(ax+b), cos(ax+b), divide by a.
- Watch the minus sign: sin goes to minus cos; cos goes to plus sin.
- Radians: trig integrals assume x is in radians — set your GDC accordingly.
The standard integrals
Special function integrals
∫ 1x dx = ln|x| + c • ∫ ex dx = ex + c
∫ sin x dx = −cos x + c • ∫ cos x dx = sin x + c
✓ all four are in the formula booklet
🤔 Why ln|x| and not the power rule?
The power rule says raise the index and divide by the new index. For 1x = x−1 the new index would be 0 — and you can’t divide by 0. So 1x is the one power that breaks the rule, and its integral is the natural log instead. The modulus |x| is there because log is only defined for positive inputs.
🧠 “Sin drops a minus, cos stays positive”
Going forwards (differentiating), cos picks up the minus: sin → cos, cos → −sin. Going backwards (integrating) the minus lands on sin instead: ∫ sin = −cos, ∫ cos = +sin. If unsure, differentiate your answer to check it comes back.
A linear function inside
When the inside is a linear expression like ax + b, you’re reversing the chain rule. Integrate as normal, then divide by a (the derivative of the inside).
Reverse chain rule (linear inside)
∫ e(ax+b) dx = 1a e(ax+b) + c
✓ extends the booklet results; AI HL examinable
Quick check: after dividing by a, differentiate mentally — the 1a and the chain-rule a should cancel to give you back the original. If they don’t, you divided the wrong way.
🧭 Recipe — integrating special functions
- Identify each term: reciprocal, exponential, sin, or cos.
- Apply the standard integral for that term (mind the sign on sin/cos).
- Linear inside? Divide that term by a, the coefficient of x.
- Add “+ c“ for an indefinite integral, or apply limits for a definite one.
Worked examples
Reciprocal → ln, exponential → itself.
3 · 1x → 3 ln|x|
ex → ex
3 ln|x| + ex + c
WE 2Find ∫ (2 cos x − 5 sin x) dx
cos → +sin, sin → −cos. The two minus signs on the sin term make a plus.
2 cos x → 2 sin x
−5 sin x → −5(−cos x) = +5 cos x
2 sin x + 5 cos x + c
Linear inside with a = 4 — integrate, then divide by 4.
inside = 4x, so a = 4
e4x → 14 e4x
14 e4x + c
WE 4Evaluate ∫13 1x dx, giving an exact answer
Integrate to ln|x|, then apply the limits as F(3) − F(1).
∫ 1x dx = ln|x|
= ln 3 − ln 1
ln 1 = 0
ln 3 (≈ 1.10)
WE 5A curve has f′(x) = 4 cos x and passes through (0, 5). Find f(x).
Integrate (cos → +sin), then substitute the point to fix c.
f(x) = 4 sin x + c
at (0, 5): 4 sin 0 + c = 5
0 + c = 5 → c = 5
f(x) = 4 sin x + 5
💡 Top tips
- Memorise the four standard integrals cold — they’re quick marks.
- Differentiate to check — your integral should differentiate back to the original.
- Divide by a whenever the inside is ax + b.
- Radians mode on the GDC for any trig integral.
- Keep the modulus: it’s ln|x|, not ln x.
- Constants pull out: ∫ 3 cos x = 3 sin x + c.
⚠ Common mistakes
- Sign slip — writing ∫ sin x = cos x instead of −cos x.
- Using the power rule on 1x — it gives ln|x|, not x00.
- Forgetting to divide by a for a linear inside.
- Multiplying by a instead of dividing — check by differentiating.
- Degrees mode on the GDC giving wrong trig values.
Next up — Integration by Substitution. You’ve now got the standard integrals for the special functions and the divide-by-a trick for a linear inside. The next topic generalises that idea: when the inside isn’t just linear, a substitution lets you swap the awkward function for a single new variable and integrate cleanly.
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