Integrate (x^3e^x^2)/(x^2+1)^2?

∫ x^3 e^(x^2) dx /(x^2+1)^2
= ∫ x x^2 e^(x^2) dx /(x^2+1)^2

Let t= x^2
dt = 2x dx
x dx = (1/2) dt

∫ x x^2 e^(x^2) dx /(x^2+1)^2 dx = (1/2) ∫ t e^t /(t+1)^2 dt

Integrate ∫ t e^t /(t+1)^2 dt by parts
dv = 1/(t+1)^2 dt ; v= -1/(t+1)
u = t e^t ; du = (e^t + t e^t) dt

∫ u dv = u v - ∫ v du

∫ t e^t /(t+1)^2 dt = - e^t t / (1+t) - ∫ ( -1/(t+1)) (e^t + t e^t) dt
∫ t e^t /(t+1)^2 dt = - e^t t / (1+t) + ∫ e^t dt /(t+1) + ∫ t e^t dt /(t+1)

∫ t e^t dt /(t+1) = ∫ (t+1-1) e^t dt /(t+1) = ∫ e^t dt - ∫ e^t dt /(t+1)
∫ t e^t /(t+1)^2 dt = - e^t t / (1+t) + ∫ e^t dt /(t+1) + [ ∫ e^t dt + ∫ e^t dt /(t+1) ]
∫ t e^t /(t+1)^2 dt = - e^t t / (1+t) + ∫ e^t dt /(t+1) + ∫ e^t dt - ∫ e^t dt /(t+1)
∫ t e^t /(t+1)^2 dt = - e^t t / (1+t) - ∫ e^t dt
∫ t e^t /(t+1)^2 dt = - e^t t / (1+t) + e^t
(1/2) ∫ t e^t /(t+1)^2 dt = - e^t t / (2(1+t)) + (1/2) e^t

replace t by x^2
= - e^(x^2) x^2 / (2(1+x^2)) + (1/2) e^(x^2)
= - e^(x^2) x^2 / (2+2x^2) + (1/2) e^(x^2) + C...Show more