Indefinite Integration is the reverse process of differentiation. Given a function f(x), we seek F(x) such that F'(x) = f(x). The result is written as integral f(x) dx = F(x) + C, where C is the arbitrary constant of integration. This topic carries 2-3 questions per year in JEE Main and is fundamental to all subsequent calculus topics.

The foundation rests on standard integrals that must be memorized: integral x^n dx = x^(n+1)/(n+1) + C (n != -1), integral 1/x dx = ln|x| + C, integral e^x dx = e^x + C, integral sin x dx = -cos x + C, integral cos x dx = sin x + C, and the inverse trigonometric forms like integral 1/(1+x^2) dx = arctan(x) + C and integral 1/sqrt(1-x^2) dx = arcsin(x) + C.

The six major techniques of integration form the core toolkit. Substitution (change of variable) is the most frequently used: when the integrand has the form f(g(x))*g'(x), substitute u = g(x). Integration by Parts uses the formula integral u dv = uv - integral v du, with the LIATE rule guiding the choice of u. Partial Fractions decomposes rational functions P(x)/Q(x) (with deg P < deg Q) into simpler fractions based on the factors of Q(x).

Trigonometric integrals use identities to simplify powers and products of trig functions. Key identities include sin^2 x = (1-cos 2x)/2, cos^2 x = (1+cos 2x)/2, and product-to-sum formulas. Trigonometric substitution handles radicals: sqrt(a^2-x^2) uses x = a sin theta, sqrt(a^2+x^2) uses x = a tan theta, and sqrt(x^2-a^2) uses x = a sec theta.

Special forms provide shortcuts: integral e^x[f(x)+f'(x)] dx = e^x*f(x) + C is heavily tested in JEE. The standard forms for 1/(x^2+a^2), 1/(x^2-a^2), 1/sqrt(x^2+a^2), and sqrt(a^2-x^2) must be memorized with their exact results.

The Weierstrass substitution t = tan(x/2) converts any rational expression in sin x and cos x into a rational function of t, using sin x = 2t/(1+t^2), cos x = (1-t^2)/(1+t^2), dx = 2dt/(1+t^2). While powerful, it often leads to complex algebra and should be used as a last resort.

Completing the square is essential for integrals involving ax^2 + bx + c in the denominator. Writing it as a(x+p)^2 + q reduces the problem to a standard form.

Common JEE patterns include: recognizing the derivative of the denominator in the numerator (giving a logarithm), splitting rational integrands strategically, and identifying the e^x[f+f'] pattern. The ability to quickly identify which technique applies to a given integrand is the key competitive advantage.