List of Formulas

(1) $∫ x^n dx = {x^{n+1}}/{n + 1} + c$

(2) $∫ 1/x dx = log|x| + c$

(3) $∫a^x dx = {a^x}/{loga} + c$

(4) $∫e^x dx = e^x + c$

(5) $∫sinx dx = – cosx + c$

(6) $∫cosx dx = sinx + c$

(7) $∫tanx dx = log |secx| + c$

(8) $∫cotx dx = log |sinx| + c$

(9) $∫secx dx = log |secx + tanx| + c$

(10) $∫cosecx dx = log |cosecx – cotx| + c$

(11) $∫sec^2 x dx = tanx + c$

(12) $∫cosec^2 x = – cotx + c$

(13) $∫secx tanx dx = secx + c$

(14) $∫cosecx cotx dx = – cosecx + c$

(15) $∫1/{√{a^2 – x^2}} dx = sin^{-1}x/a + c $

(16) $∫1/{a^2 + x^2} dx = 1/a tan^{-1}x/a + c$

(17) $∫{1}/{x√{x^2 – 1}} dx = sec^{-1}x + c = – cosec^{-1}x + c$

(18) $∫1/{√{a^2 – x^2}} dx = sin^{-1}(x/a) + c$

(19) $∫1/{x√{x^2 – a^2}} dx = 1/a sec^{-1} (x/a) + c$

(20) $∫{dx}/{√{x^2 + a^2}} dx = log(x + √{x^2 + a^2}) + c$ OR $sinh^{-1}x/a + c$

(21) $∫{dx}/{a^2 – x^2} dx = 1/{2a} log({a+x}/{a-x}) + c$ OR $1/a tanh^{-1} x/a + c$

(22) $∫1/{x^2 – a^2} dx = 1/{2a} log {x-a}/{x+a} + c$

(23) $∫√{a^2 – x^2} dx = x/a √{a^2 – x^2} + a^2/2 sin^{-1} (x/a) + c $

(24) $∫√{x^2 – a^2} dx = x/a √{x^2 – a^2} – a^2/2 log [x + √{x^2 – a^2}] + c$

(25) $∫√{x^2 + a^2} dx = x/a √{x^2 + a^2} + a^2/2 log [x + √{x^2 + a^2}] + c$

(26) $∫e^x [f(x) + f'(x)] dx = e^x f(x)$

(27) $∫e^{ax}$ sinbx dx = ${e^{ax}}/{a^2 + b^2} (a sinbx – b cosbx)$

(28) $∫ e^{ax}$cosbx dx = ${e^{ax}}/{a^2 + b^2} (a cosbx + b sinbx)$

(29) $∫1/{√{x^2 – a^2}} dx = log (x + √{x^2 – a^2}) + c$

(30) $∫u . v dx = u ∫ v dx – ∫[{du}/{dx} . ∫ v dx] dx + c$

(31) $∫ [f(x)]^n f’ (x) dx = {[f(x)]^{n+1}}/{n+1}$ , n ≠ -1

(32) $∫ {f'(x)}/{f(x)} dx = log f(x)$

(33) $∫ e^{f(x)} f'(x) dx = e^{f(x)}$

(34) $∫sin (f(x)) f'(x) dx = – cos (f(x))$

(35) $∫ cos (f(x)) f'(x) dx = sin (f(x))$

(36) $∫{f'(x)}/{√{f(x)}} dx = 2 √{f(x)}$

(37) $∫√{f(x)} f'(x) dx = 2/3 [f(x)]^{3/2}$

(38) ${∫↙0↖a} f(x) dx = {∫↙0↖a} f(a-x) dx$

(39) $∫↙a↖b f(x) dx = ∫↙a↖c f(x) dx + ∫↙c↖b f(x) dx$ a

(40) $∫↙0↖{2a} f(x) dx = ∫↙0↖a f(x) dx + ∫↙0↖a f(2a – x) dx$

(41) $\table ∫↙a↖{-a} f(x) dx, =, 2 ∫↙0↖a f(x) dx, \text"if f(x) is even";,=, 0 , \text"if f(x) is odd"$