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^$\frac{n+1}{(n+1)}$ + 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 $\frac{1}{1+x^2}$ 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$\frac{x}{Q}$(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 = $\frac{1-cos 2x}{2}$, $cos^2$ x = $\frac{1+cos 2x}{2}$, 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 $\frac{1}{x^2+a^2}$, $\frac{1}{x^2-a^2}$, 1/sqrt(x^{2+a}^2), and sqrt(a^{2-x}^2) must be memorized with their exact results.

The Weierstrass substitution t = tan$\frac{x}{2}$ converts any rational expression in sin x and cos x into a rational function of t, using sin x = 2$\frac{t}{1+t^2}$, cos x = $\frac{1-t^2}{(1+t^2)}$, dx = 2$\frac{dt}{1+t^2}$. 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.