Pattern 1: e^x[f(x) + f'(x)] Recognition (Very High Frequency) JEE frequently disguises this pattern. Example: integral e^x(x+1)/(x+2)^2 dx. Write (x+1)/(x+2)^2 = 1/(x+2) - 1/(x+2)^2. With f(x) = 1/(x+2), f'(x) = -1/(x+2)^2, answer is e^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 (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 1/(quadratic) or 1/sqrt(quadratic). Complete the square and apply the standard result.

Pattern 5: Trig Integrals with Smart Substitution integral sqrt(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 dx/(x(x^5+1)) — multiply by x^4/x^4 to create x^5 in both places, then substitute t = x^5.