JEE Maths — Indefinite Integration

Chapter Overview & Weightage

Indefinite Integration is one of the heaviest scoring chapters in JEE Mains and Advanced. The chapter trains pattern recognition — recognising which substitution or partial-fraction structure unlocks an integral within seconds.

Typical JEE weightage: $2-3$ questions in JEE Main, often $1-2$ in JEE Advanced (may be combined with definite integration or differential equations).

Year	JEE Main Qs	JEE Advanced Qs
2020	2	1
2021	3	2
2022	2	1
2023	3	2
2024	3	2

Key Concepts You Must Know

Standard integrals (memorise the table of 25 forms cold)
Substitution: trigonometric, algebraic ( $x \pm 1/x$ , $1/x$ swap)
Integration by parts: ILATE rule for choice of $u$
Partial fractions for rational functions
Trigonometric integrals: $\int \sin^m x \cos^n x \, dx$
Reduction formulas (especially for $\int \sin^n x\,dx$ , $\int \cos^n x\,dx$ )
Special integrals like $\int \sqrt{a^2 - x^2}\,dx$ , $\int dx/(a^2 + x^2)$

Important Formulas

$\int u\,dv = uv - \int v\,du$

ILATE: choose $u$ from Inverse trig → Logarithmic → Algebraic → Trigonometric → Exponential.

$\int \frac{dx}{a^2 + x^2} = \frac{1}{a}\tan^{-1}(x/a) + C$

$\int \frac{dx}{a^2 - x^2} = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C$

$\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(x/a) + C$

For integrands like $(x^2 \pm 1)/(x^4 + kx^2 + 1)$ , divide top and bottom by $x^2$ , then substitute $u = x \mp 1/x$ .

Solved Previous Year Questions

PYQ 1 (JEE Main 2023)

Evaluate $\int \frac{\sin^{-1} x}{\sqrt{1 - x^2}}\,dx$ .

Substitute $u = \sin^{-1}x$ , $du = dx/\sqrt{1-x^2}$ . Integral becomes $\int u\,du = u^2/2 + C = (\sin^{-1}x)^2/2 + C$ .

PYQ 2 (JEE Main 2024)

$\int \frac{1}{x(x^5 + 1)}\,dx$ .

Multiply num and denom by $x^4$ : $\int \frac{x^4}{x^5(x^5+1)}\,dx$ . Let $u = x^5$ , $du = 5x^4 dx$ . Integral = $\tfrac{1}{5}\int \frac{du}{u(u+1)} = \tfrac{1}{5}\int(1/u - 1/(u+1))du = \tfrac{1}{5}\ln|u/(u+1)| + C = \tfrac{1}{5}\ln|x^5/(x^5+1)| + C$ .

PYQ 3 (JEE Advanced 2022)

$\int e^x(\sin x + \cos x)\,dx$ .

Recognise $f(x) + f'(x)$ pattern with $f(x) = \sin x$ . Standard result: $\int e^x(f(x) + f'(x))\,dx = e^x f(x) + C$ .

So integral = $e^x \sin x + C$ .

Difficulty Distribution

Difficulty	% of JEE Qs	Typical type
Easy	$30\%$	Direct standard integrals
Medium	$50\%$	Substitution + by parts combos
Hard	$20\%$	Tricky pattern recognition (e.g., $e^x(f + f')$ )

Expert Strategy

Maintain a “weird integrals notebook” — write down every non-trivial pattern you encounter. By the time you finish 200 problems, you’ll have seen most patterns JEE uses.

For products like $e^x \cdot (\text{trig})$ , look for $f(x) + f'(x)$ form. The shortcut $\int e^x(f + f')dx = e^x f$ saves 2 minutes per question.

For symmetric rational integrands, try the $x \pm 1/x$ substitution first. JEE loves this pattern — it appears almost yearly.

Common Traps

Forgetting the constant of integration. JEE Main and Advanced will mark a ”+ C”-less answer wrong even if the antiderivative is otherwise correct.

Using $\int (1/x)dx = \ln x$ without absolute value. The correct form is $\ln|x| + C$ . JEE Advanced has docked marks for this.

Choosing $u$ wrong in ILATE. The “I” stands for inverse trig; common mistake is to choose the trig part instead. Pick the function highest in ILATE for $u$ .

Ignoring the chain rule when substituting back. After computing $\int u\,du$ , replace $u$ with the original expression, not just leave it as $u$ .