*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[$e^x$*f(x)] = $e^x$*f(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$$\frac{(1+x}{(1+x^2)} + (1-x)^$\frac{2}{1+x^2}$^2) dx: need to identify f and f' carefully
• integral $e^x$($\frac{1}{1+x^2}$ - 2$\frac{x}{1+x^2}$^2) dx = e^$\frac{x}{1+x^2}$ + C [f = $\frac{1}{1+x^2}$]

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

Formula 3: Linear/Quadratic Split integral $\frac{px+q}{(ax^2+bx+c)}$ dx: Write px+q = $\frac{p}{2a}$(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 + $\frac{(n-1}{n}$)*integral sin^(n-2)(x) dx
• integral $cos^n$(x) dx = (cos^(n-1)(x)*sin(x))/n + $\frac{(n-1}{n}$)*integral cos^(n-2)(x) dx
• integral $x^n$*e^(ax) dx = ($x^n$*e^(ax))/a - $\frac{n}{a}$*integral x^(n-1)*e^(ax) dx