Mistake 1: Forgetting + C Every indefinite integral must include + C. Losing marks in JEE for this is unforgivable. In verification (differentiation check), C disappears, so it's easy to forget.

Mistake 2: Incorrect substitution differential If u = $x^2$, then du = 2x dx, NOT du = dx. Always compute du correctly. Wrong: integral sin($x^2$) dx "=" -cos($x^2$) + C (this is WRONG — there's no 2x factor)

Mistake 3: Partial fractions without long division integral $\frac{x^3+1}{(x^2-1)}$ dx: degree of numerator (3) >= degree of denominator (2). MUST divide first: $x^{3+1}$ = ($x^{2-1}$)*x + (x+1). So integral = integral x dx + integral $\frac{x+1}{(x^2-1)}$ dx = $x^2$/2 + integral $\frac{1}{x-1}$ dx = $x^2$/2 + ln|x-1| + C

Mistake 4: Sign errors in by parts integral u dv = uv - integral v du (note the MINUS sign). Common error: writing + instead of - in the second term.

Mistake 5: Wrong LIATE choice integral x*arctan(x) dx: u should be arctan(x) (I before A in LIATE), not x. Choosing u = x leads to a more complicated integral.

Mistake 6: Missing absolute value in logarithms integral $\frac{1}{x-3}$ dx = ln|x-3| + C, NOT ln(x-3) + C. The absolute value is essential when x-3 can be negative.

Mistake 7: Incorrect trig substitution back-substitution After x = a*sin(t), you get an answer in terms of t. Must convert back to x using: t = arcsin$\frac{x}{a}$, cos(t) = sqrt$\frac{a^2-x^2}{a}$, etc.

Mistake 8: Applying $e^x$[f+f'] formula incorrectly integral $e^x$(x + 1/x) dx: Is 1/x the derivative of x? d/dx(x) = 1, not 1/x. So this is NOT the $e^x$[f+f'] form. Don't force the pattern.