INTEGRATION FORMULAS

∫ 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)

∫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

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

∫ e^ax cos bx dx =  e^ax/(a² + b²) (a cos bx + b sin bx) + c

∫ 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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