Pattern 1: $e^x$[f(x) + f'(x)] Recognition (Very High Frequency) JEE frequently disguises this pattern. Example: integral e^x$$\frac{x+1}{(x+2)}^2 dx. Write $\frac{x+1}{(x+2)}$^2 = $\frac{1}{x+2}$ - $\frac{1}{x+2}$^2. With f(x) = $\frac{1}{x+2}$, f'(x) = -$\frac{1}{x+2}$^2, answer is e^$\frac{x}{x+2}$ + C.

Pattern 2: Derivative of Denominator in Numerator (High Frequency) If numerator = k * d/dx(denominator), answer is k*ln|denominator| + C. Always check this first for rational integrands.

Pattern 3: Splitting the Numerator For $\frac{px+q}{(ax^2+bx+c)}$, write px+q = L*(2ax+b) + M. The first part gives a log, the second gives an arctan after completing the square.

Pattern 4: Standard Forms with Completing the Square Nearly every JEE paper has an integral of the form $\frac{1}{quadratic}$ or 1/sqrt(quadratic). Complete the square and apply the standard result.

Pattern 5: Trig Integrals with Smart Substitution integral sqrt$\frac{tanx}{sin2x}$ dx type problems: multiply by $sec^{2x}$/$sec^{2x}$ and substitute t = tanx or t = sqrt(tanx).

Pattern 6: Reduction to Known Forms Multiply numerator and denominator by strategic factors. Example: integral $\frac{dx}{x(x^5+1}$) — multiply by $x^4$/$x^4$ to create $x^5$ in both places, then substitute t = $x^5$.