The chain rule differentiates composite functions: the derivative of sin(x2) is cos(x2)⋅2x — differentiate the outer function, then multiply by the derivative of the inside. Integration asks the reverse: given cos(x2)⋅2x, find the original function. That's what u substitution does — it reverses the chain rule.
The Core Intuition
When an integral looks like "a function times the derivative of its inside," substitute the inside with u and the derivative of the inside with du. Consider:
∫2xcos(x2)dx
Here 2x is the derivative of x2, and x2 is the inside of cos(x2). Let u=x2, so du=2xdx:
∫cos(u)du=sin(u)+C=sin(x2)+C
Check: the derivative of sin(x2) is cos(x2)⋅2x.
The Precise Statement
∫f(g(x))⋅g′(x)dx=∫f(u)duwhere u=g(x),du=g′(x)dx
Valid provided g is differentiable and the resulting integral in u is simpler.
The Step-by-Step Method
Identify a function g(x) whose derivative g′(x) also appears (possibly up to a constant factor).
Setu=g(x), compute du=g′(x)dx.
Rewrite the entire integral in u and du — every x and dx must be replaced.
Integrate with respect to u.
Substitute backu=g(x).
Watch out
You cannot mix variables. If any x remains after substitution, you chose the wrong u (or must solve for x in terms of u — rare).
A Second Example (with a constant factor)
Evaluate ∫xx2+1dx. Let u=x2+1, so xdx=21du:
∫u⋅21du=21⋅32u3/2+C=31(x2+1)3/2+C
When Does It Work?
When the integrand is something times the derivative of something inside. Common patterns:
The key idea is to rewrite the integrand using the half-angle identity 1+cosx1−cosx=tan22x, then integrate tan2u=sec2u−1 to get 2tan2x−x+C.
Why this approach works
When you see a ratio of 1±cosx, your first instinct should be half-angle formulas. They exist precisely to simplify such expressions. The identity cosx=2cos22x−1=1−2sin22x lets us rewrite both numerator and denominator in terms of 2x, and the ratio collapses beautifully into a single squared tangent.
Why tangent? Because 1+cosx1−cosx is a classic form for tan22x. Once you have tan2u, you integrate it by recalling that tan2u=sec2u−1 — and sec2u integrates to tanu, while 1 integrates to u. The whole thing becomes a clean, two-step process.
Step-by-step solution
1. Apply the half-angle identity.
We use cosx=1−2sin22x=2cos22x−1. Then:
1−cosx=1−(1−2sin22x)=2sin22x
1+cosx=1+(2cos22x−1)=2cos22x
So the integrand becomes:
1+cosx1−cosx=2cos22x2sin22x=tan22x
Tip
You can also derive this directly from the identity tan22x=1+cosx1−cosx — it's worth memorising as a time-saver in exams.
Mistake 1: Forgetting the factor 2 from dx=2du when substituting u=x/2.
Why it's wrong: integrating sec2(x/2) gives 2tan(x/2), not tan(x/2), because the argument's slope is 21. Correct approach: ∫sec22xdx=2tan2x, yielding 2tan2x−x+C.
Mistake 2: Cancelling 1+cosx1−cosx to a wrong simplification. …
The integral ∫9−4x2dx is a standard inverse sine form. By rewriting the denominator as 4(49−x2) and using substitution u=2x, we get the result 21sin−1(32x)+C.
When you see a square root with a constant minus a square term, your mind should immediately jump to the inverse trigonometric integrals. The classic formula is:
∫a2−u2du=sin−1(au)+C
Our job is to force the given integral into this exact shape. The denominator is 9−4x2. Notice that 9=32, so we have a=3 in the formula. But the 4x2 term is not a pure u2 — it has a coefficient 4. That’s the only obstacle.
The key insight: Factor out the 4 from inside the square root. Write:
9−4x2=4(49−x2)=249−x2
Now the integral becomes:
∫249−x2dx=21∫(23)2−x2dx
This is exactly the inverse sine form with a=23 and u=x. So:
21sin−1(3/2x)+C=21sin−1(32x)+C
That’s the answer. But let’s walk through it step by step with a substitution to make it foolproof.
Identify the target form. We want ∫a2−u2du. Here, the denominator has 9−4x2. Compare with a2−u2: we need a2=9 and u2=4x2. So set u=2x. Then du=2dx, so dx=2du.
Q.If ∫b2+c2x23axdx=Alog∣b2+c2x2∣+K, then the value of A is:
(A) 3a
(B) 2b23a
(C) b2c23a
(D) 2c23a
›Reveal solutionSolution
The integral fits the pattern ∫udu=log∣u∣+C after a substitution. The constant A turns out to be 2c23a, which corresponds to option (D).
The problem gives you the result of an integral and asks you to identify the constant A that makes the equation true. This is a classic "match the form" question — you don't need to guess; you just need to perform the integration carefully and compare.
The key insight is that the integrand b2+c2x23ax is a rational function where the numerator is almost the derivative of the denominator. The derivative of b2+c2x2 is 2c2x. Our numerator is 3ax, which is a constant multiple of x. So a simple substitution u=b2+c2x2 will turn the integral into ∫udu.
Let's work through it step by step.
Set up the substitution.
Let u=b2+c2x2. Then du=2c2xdx, so xdx=2c2du.
Rewrite the integral in terms of u.
The integral is ∫b2+c2x23axdx=∫u3a⋅(xdx).
Substitute xdx=2c2du:
∫u3a⋅2c2du=2c23a∫udu.
Integrate.∫udu=log∣u∣+C, so
2c23alog∣u∣+C=2c23alog∣b2+c2x2∣+K,
where K is the constant of integration (we renamed C to K to match the problem).