Shortcut 1: Differentiation Check Unsure of your answer? Differentiate it. If you get back the original integrand, you're correct. This takes 30 seconds and catches sign/coefficient errors.

Shortcut 2: Recognizing d/dx in Numerator Before any technique, check if the numerator is the derivative of the denominator (gives ln) or of the expression under a root. This solves ~30% of JEE integration problems instantly.

Shortcut 3: Standard Form Recognition Memorize these on sight:

• $\frac{1}{x^2+a^2}$ -> $\frac{1}{a}$arctan$\frac{x}{a}$
• $\frac{1}{x^2-a^2}$ -> $\frac{1}{2a}$ln|$\frac{x-a}{(x+a)}$|
• 1/sqrt(a^{2-x}^2) -> arcsin$\frac{x}{a}$
• 1/sqrt(x^{2+a}^2) -> ln|x+sqrt(x^{2+a}^2)|

Shortcut 4: Splitting Linear Numerator Over Quadratic For integral $\frac{px+q}{(ax^2+bx+c)}$ dx: always split as px+q = lambda*(derivative of denominator) + mu This converts to ln + arctan immediately.

Shortcut 5: Trig Power Quick Results

• integral $sin^2$(x) dx = x/2 - sin$\frac{2x}{4}$ + C
• integral $cos^2$(x) dx = x/2 + sin$\frac{2x}{4}$ + C
• integral $sin^3$(x) dx = -cos(x) + cos^3$$\frac{x}{3} + C
• integral $cos^3$(x) dx = sin(x) - sin^3$$\frac{x}{3} + C

Shortcut 6: Use Options in MCQ If stuck, differentiate each option. The one whose derivative matches the integrand is correct. This is a valid strategy for difficult integrals in MCQ format.

Shortcut 7: integral f'$\frac{x}{sqrt}$(f(x)) dx = 2*sqrt(f(x)) + C Direct substitution result — saves time on many problems.

Shortcut 8: For integral $\frac{dx}{a*sin(x}$+b*cos(x)) Write asin(x)+bcos(x) = sqrt(a^{2+b}^2)*sin(x+alpha). integral = integral csc$\frac{x+alpha}{sqrt}$(a^{2+b}^2) dx = standard csc integral.