Formula 1: e^x[f(x) + f'(x)] = e^x*f(x) + C This is the most tested special form in JEE. Proof: d/dx[e^xf(x)] = e^xf(x) + e^x*f'(x) = e^x[f(x) + f'(x)].

Examples:

• integral e^x(sin(x) + cos(x)) dx = e^x*sin(x) + C [f = sin(x)]
• integral e^x(1/x + ln(x)) dx = e^x*ln(x) + C [f = ln(x)]
• integral e^x((1+x)/(1+x^2) + (1-x)^2/(1+x^2)^2) dx: need to identify f and f' carefully
• integral e^x(1/(1+x^2) - 2x/(1+x^2)^2) dx = e^x/(1+x^2) + C [f = 1/(1+x^2)]

Formula 2: Completing the Square For integral 1/(ax^2+bx+c) dx or integral 1/sqrt(ax^2+bx+c) dx: Write ax^2+bx+c = a[(x+b/2a)^2 + (c/a - b^2/4a^2)] Then apply arctan or log/arcsin standard forms.

Formula 3: Linear/Quadratic Split integral (px+q)/(ax^2+bx+c) dx: Write px+q = (p/2a)(2ax+b) + (q - pb/2a) First part gives log, second part gives arctan (after completing the square).

Formula 4: integral sqrt(ax^2+bx+c) dx Complete the square to get sqrt((x+alpha)^2 +/- beta^2) form, then use standard results.

Formula 5: Reduction Formulas

• integral sin^n(x) dx = -(sin^(n-1)(x)*cos(x))/n + ((n-1)/n)*integral sin^(n-2)(x) dx
• integral cos^n(x) dx = (cos^(n-1)(x)*sin(x))/n + ((n-1)/n)*integral cos^(n-2)(x) dx
• integral x^ne^(ax) dx = (x^ne^(ax))/a - (n/a)*integral x^(n-1)*e^(ax) dx