integration jee notes boards

Integration Formulas

Indefinite Integration:

1. If f and g are functions of x such that g’(x) = f(x) then,

Here, c is called the constant of integration.

2. Standard formula:

(c) ∫e^x dx = e^x+c

(e) ∫ sin x dx  = -cos x + c

(f) ∫ cos x dx  = sin x + c

(g) ∫ sec² x dx  = tan x + c

(h) ∫ cosec² x dx  = -cot x + c

(h) ∫ sec x tan x dx = sec x + c

(i) ∫ cosec x cot x dx = – cosec x + c

(j) ∫ cot x dx = log|sin x| + c 

(k) ∫ tan x dx = -log|cos x| + c

(l) ∫ sec x dx = log |sec x + tan x| + c

(m) ∫ cosec x dx = log |cosec x – cot x| + c

If in place of x we have (ax+b), then the same formula is applicable but we must divide by coefficient of x or derivative of (ax+b), i.e., a.

(iii) ∫ e^ax+b dx = (1/a) e^ax+b+c

(v)  ∫ sin(ax + b)dx = -(1/a) cos(ax + b) + c

(vi) ∫ cos(ax + b) dx = (1/a) sin(ax + b) + c

(vii)  ∫ tan(ax + b)dx = (1/a) ln sec(ax + b) + c

(viii) ∫ cot(ax+b)dx = (1/a) ln sin (ax + b) + c

(ix) ∫ sec²(ax+b)dx = (1/a) tan(ax+b) + c

(x) ∫ cosec² (ax+b) dx = -(1/a) cot (ax+b) + c

(xi) ∫ sec (ax+b) ⋅ tan (ax+b) dx = (1/a) sec (ax+b) + c

(xii) ∫ cosec (ax+b) ⋅ cot (ax+b) dx = -(1/a) cosec (ax+b) + c

(xiii) ∫ sec x dx = ln (sec x + tan x) + c  or 

(xiv) ∫ cosec x dx = ln (cosec x – cot x) + c  or  ln tan (x/2) +c  or –ln (cosec x + cot x)+ c