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:

• 1/(x^2+a^2) -> (1/a)arctan(x/a)
• 1/(x^2-a^2) -> (1/2a)ln|(x-a)/(x+a)|
• 1/sqrt(a^2-x^2) -> arcsin(x/a)
• 1/sqrt(x^2+a^2) -> ln|x+sqrt(x^2+a^2)|

Shortcut 4: Splitting Linear Numerator Over Quadratic For integral (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(2x)/4 + C
• integral cos^2(x) dx = x/2 + sin(2x)/4 + C
• integral sin^3(x) dx = -cos(x) + cos^3(x)/3 + C
• integral cos^3(x) dx = sin(x) - sin^3(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'(x)/sqrt(f(x)) dx = 2*sqrt(f(x)) + C Direct substitution result — saves time on many problems.

Shortcut 8: For integral dx/(asin(x)+bcos(x)) Write asin(x)+bcos(x) = sqrt(a^2+b^2)*sin(x+alpha). integral = integral csc(x+alpha)/sqrt(a^2+b^2) dx = standard csc integral.