INTEGRATION FORMULAS

Basic Formulas in Integration

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

∫ (1/xⁿ) dx = -1/(n - 1) x⁽ⁿ⁻¹⁾  + c

∫ (1/x) dx = log x  + c

∫ e^(x) dx = e^x  + c

∫ a^(x) dx = a^x/(log a)  + c

∫ sin x dx = - Cos x + c

∫ cos x dx = Sin x + c 

∫ cosec ² x dx = - Cot x  + c

∫ sec² x dx = tan x  + c

∫ sec x tan x dx = sec x  + c

∫ cosec x cot x dx = - cosec x  + c

∫ 1/(1 + x²) dx =  tan ⁻ ¹x  + c

∫ 1/ √(1 - x²) dx = sin ⁻ ¹x  + c

∫ tan x dx = log |sec x| + c

∫cot x dx = log |sin x| + c

∫cosec x dx = log |cosec x - cot x| + c

∫sec x dx = log |sec x + tan x| + c

∫u dv = uv - ∫vdu

Bernoulli formula :

∫u dv = uv - u'v1 + u''v2 - ............ 

∫f'(x)/f(x)  dx  =  log |x| + c

∫ f'(x) [f(x)]n dx  =  [f(x)]n + 1/(n + 1)


Integration Formula When x is in the Form ax + b

∫e(a x + b) dx  =   (1/a)e(a x + b) + c

∫(ax + b)ⁿdx  =  (1/a)(ax + b)(n + 1)/(n + 1)   + c

∫1/(ax + b)dx  =  (1/a)log(ax + b) + c

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

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

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

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

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

∫sec(ax + b)tan (ax + b)dx  =  sec(ax + b) + c

∫1/[1 + (ax)2]dx  =  (1/a)tan-1(ax) + c

∫1/ √[1 - (ax)2] dx  =  (1/a) sin-1(ax)   + c

Integration Formula in the form e^ax sin bx or e^ax cos bx

∫ eax sin bx dx  =  eax/(a2 + b2) (a sin bx - b cos bx) + c

∫ eax cos bx dx =  eax/(a2 + b2) (a cos bx + b sin bx) + c

Standard Integration Formulas

∫ dx/(a2-x2)  =  (1/2a) log [(a + x)/(a - x)] + c

∫ dx/(x2-a2)  =  (1/2a) log [(x - a)/(x + a)] + c

∫ dx/(a2+ x2)  =  (1/a) tan-1 (x/a) + c

∫ dx/(a2- x2)  =  sin-1 (x/a) + c

∫ dx/(√x2- a2)  =  log (x + (√x2- a2)) + c

∫ dx/(√x2+ a2)  =  log (x + (√x2+ a2)) + c

√(a2- x2)  =  (x/2) √(a2- x2) + (a2/2) sin-1(x/a) + c

∫ √(x2- a2)  =  (x/2) √(x2- a2) - (a2/2) log(x+√(x2- a2))+c

∫ √(x2+a2)  =  (x/2) √(x2+ a2) + (a2/2) log(x+√(x2+ a2))+c

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