JEE Maths›Definite & Indefinite Integrals›Common Mistakes
Common Mistakes

Traps in Definite & Indefinite Integrals

5 mistake patterns students fall for. 3 high-frequency traps appear in almost every exam.

Forgetting to change limits after substitution

Very CommonFORMULA

When using substitution in definite integrals, the limits must change to the new variable.

Why: Students substitute the integrand correctly but forget that limits are in terms of the old variable.

WRONG: Substituting t=sin⁡xt = \sin x but keeping limits as 00 to π/2\pi/2 instead of converting to 00 to 11
RIGHT: When t=sin⁡xt = \sin x: x=0⇒t=0x = 0 \Rightarrow t = 0, x=π/2⇒t=1x = \pi/2 \Rightarrow t = 1. Use new limits with new variable.
See pattern: Substitution-Based Integrals →

Wrong application of King's property

Very CommonFORMULA

King's property: ∫abf(x) dx=∫abf(a+b−x) dx\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx. The replacement is x→(a+b−x)x \to (a+b-x), not (b−a−x)(b-a-x).

Why: Students confuse the substitution: the sum of limits (a+b)(a+b) replaces xx, not the difference.

WRONG: Replacing xx with (b−x)(b-x) when limits are aa to bb
RIGHT: Replace xx with (a+b−x)(a+b-x). For 00 to π\pi: replace with π−x\pi - x. For 00 to 11: replace with 1−x1-x.
See pattern: Properties of Definite Integrals →

Sign error in integration by parts

Very CommonSIGN ERROR

∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du. The minus sign before the second integral is often lost.

Why: The negative sign is easy to drop when the second integral itself produces another negative.

WRONG: ∫x⋅ex dx=x⋅ex+∫ex dx=xex+ex+C\int x \cdot e^x\,dx = x \cdot e^x + \int e^x\,dx = xe^x + e^x + C
RIGHT: ∫x⋅ex dx=x⋅ex−∫ex dx=xex−ex+C=ex(x−1)+C\int x \cdot e^x\,dx = x \cdot e^x - \int e^x\,dx = xe^x - e^x + C = e^x(x-1) + C
See pattern: Integration by Parts →

Missing constant of integration

CommonFORMULA

Every indefinite integral must have +C+ C. Forgetting it in MCQs can lead to wrong option.

Why: Students focus on the antiderivative and forget that indefinite integrals have a family of solutions.

WRONG: ∫2x dx=x2\int 2x\,dx = x^2 (without +C+C)
RIGHT: ∫2x dx=x2+C\int 2x\,dx = x^2 + C. In MCQs, check if options differ by a constant.

Leibniz rule: forgetting chain rule

CommonFORMULA

ddx∫0g(x)f(t) dt=f(g(x))⋅g′(x)\frac{d}{dx} \int_0^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x), not just f(g(x))f(g(x)).

Why: Students apply the fundamental theorem but forget that the upper limit is a function of xx.

WRONG: ddx∫0x2sin⁡(t) dt=sin⁡(x2)\frac{d}{dx} \int_0^{x^2} \sin(t)\,dt = \sin(x^2)
RIGHT: ddx∫0x2sin⁡(t) dt=sin⁡(x2)⋅2x\frac{d}{dx} \int_0^{x^2} \sin(t)\,dt = \sin(x^2) \cdot 2x. The g′(x)g'(x) factor is essential.
See pattern: Functional Equation + Integral →
Test yourself

Can you spot these traps under time pressure?

Take a timed quiz on Definite & Indefinite Integrals and see if you avoid the mistakes above.

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