On this page
Substitution is the first integration method you learn and the one you use most, often inside parts or partial fractions. The algebra is easy. The hard part is choosing t, and a poor choice leaves you with a mix of x and t that goes nowhere. This post gives five patterns that cover most JEE substitutions, worked examples for each, and a check that catches errors before the answer key does.
What does NCERT say about choosing t?
The JEE (Main) 2026 syllabus lists integration by substitution in Unit 8. Section 7.3.1 of the current NCERT Class 12 chapter on integrals gives the method: put x = g(t), so dx = g′(t) dt, and the integral becomes one in t. It then gives the rule that matters: "Usually, we make a substitution for a function whose derivative also occurs in the integrand."
So the question to ask is not "what looks complicated?" but "which piece has its derivative sitting nearby?"
Which five patterns cover most substitutions?
Inner function, derivative outside. In ∫ 2x cos(x²) dx, the derivative of x² is 2x. Put t = x².
Derivative over function. In ∫ f′(x)/f(x) dx, put t = f(x); the answer is log|f(x)| + C. For example, ∫ cot x dx = log|sin x| + C.
Linear inside. In ∫ (3x + 2)⁵ dx, put t = 3x + 2, so dx = dt/3. The answer is (3x + 2)⁶/18 + C.
Root of a linear expression. In ∫ x√(x + 1) dx, put t = x + 1. The x left over becomes t − 1.
Inverse trig or log inside. In ∫ e^(tan⁻¹ x)/(1 + x²) dx, the derivative of tan⁻¹ x is 1/(1 + x²). Put t = tan⁻¹ x and the answer is e^(tan⁻¹ x) + C.
Worked example: a leftover x
Find ∫ x√(x + 1) dx.
Put t = x + 1, so dx = dt and x = t − 1. The integral becomes ∫ (t − 1)√t dt = ∫ (t^(3/2) − t^(1/2)) dt.
Integrate: (2/5)t^(5/2) − (2/3)t^(3/2) + C.
Back-substitute: (2/5)(x + 1)^(5/2) − (2/3)(x + 1)^(3/2) + C.
The key step was expressing the leftover x in terms of t. Students often stop at ∫ x√t dt, which mixes variables and cannot be integrated.
Check. Differentiate: (x + 1)^(3/2) − (x + 1)^(1/2) = (x + 1)^(1/2)(x + 1 − 1) = x√(x + 1). Correct.
Worked example: change the limits
Find ∫ from 0 to 1 of x e^(x²) dx.
Put t = x², so dt = 2x dx and x dx = dt/2. When x = 0, t = 0; when x = 1, t = 1.
The integral becomes (1/2) ∫ from 0 to 1 of eᵗ dt = (1/2)(e − 1).
You never return to x. Section 7.9 of the NCERT chapter does definite integrals by substitution this way.
When does substitution not work?
If the derivative of your chosen t is not in the integrand, even up to a constant, substitution usually fails. ∫ e^(x²) dx has no 2x outside, and putting t = x² leaves dx = dt/(2√t), which is no easier. This integral has no answer in elementary functions at all. In JEE, a definite integral like that is a signal to use a property instead; our post on definite integral properties shows how.
When the integrand is a product of two unrelated functions, such as x eˣ, substitution will not help either. That is a job for integration by parts.
Which mistakes cost the most marks?
Forgetting to convert dx. In ∫ cos 3x dx, dx = dt/3, so the answer is (sin 3x)/3 + C, not sin 3x + C.
Leaving an x behind, as in the first worked example.
Not changing the limits, or changing them and then also back-substituting.
Dropping the modulus in log|f(x)|.
Not checking. Differentiating your answer takes under a minute.
Why do worked examples help, and where do they stop?
Sweller and Cooper (1985) found that students who studied worked algebra examples went on to solve similar problems faster and with fewer errors. They also found that this benefit was specific to problems identical in structure to the examples. For substitution, that means reading solutions is a good start but will not teach you to choose t on an unfamiliar integral. You need to practise the choice itself, on mixed problems.
A practice routine
Name the pattern (10 minutes, three times a week). Take ten integrals and write only the substitution and which of the five patterns it fits. Do not solve them.
Solve and check. Solve five of them fully and differentiate each answer.
One definite integral a day, changing the limits each time.
Mix methods. Once a week, do a set that mixes substitution, parts and partial fractions. In Rohrer and Taylor's 2007 experiment, mixed practice gave "vastly superior" results a week later compared with practising one type at a time.
If you cannot see which piece to call t, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning behind each choice rather than printing an answer.
Tonight, take the integrals in NCERT Exercise 7.2 and write the substitution for each before you solve any of them.
