∫cos x / ( 1 + (sinx)^2 ) dx =

∫cos x / ( 1 + (sinx)^2 ) dx = = ∫d(sinx)/[1 + (sinx)^2] = arctan(sinx) + C[arctan（）,反正切函数啊]∫x^3 / ( 1 + x^4 ) dx = =(1/4)∫d(1 + x^4)/(1 + x^4) = ln(1 + x^4)/4 + C ∫(sec x)^3 * tan x dx = =∫(secx)^2d(secx) = (secx)^3/3 + C ∫x^2 * e^(-2x) dx = = (-1/2)∫x^2d(e^(-2x)) = -1/2e^(-2x)x^2 + ∫e^(-2x)xdx = -1/2e^(-2x)x^2 - 1/2∫xd(e^(-2x)) = -1/2e^(-2x)x^2 - 1/2e^(-2x)x + ∫e^(-2x)dx = -1/2e^(-2x)[x^2 + x + 1] + C ∫x * cos(2x) dx = = (1/2)∫xdsin(2x) = 1/2xsin(2x) - 1/2∫sin(2x)dx = 1/2xsin(2x) + 1/4cos(2x) + C ∫(cos 2x)^2 dx = = (1/2)∫[1 + cos(4x)]dx = 1/2[x + sin(4x)/4] + C ∫(13x - 6) / ( x (x-2)(x+3) )dx = ∫[1/x + 2/(x-2) + 3/(x+3)]dx = ln|x| + 2ln|x-2| + 3ln|x+3| + C ∫(x^2 + 2x - 2) / ((x-2)(x+1)) dx = = ∫[x^2 - x - 2 + 3x]dx/[(x-2)(x+1)] = ∫[1 + 3x/[(x-2)(x+1)]]dx = x + ∫[2/(x-2) + 1/(x+1)]dx = x + 2ln|x-2| + ln|x+1| + C