(c) find the integral of e^x/ e^3x + e^x dx
e^x/ (e^3x + e^x) dx
Let u=e^x, du = e^x dx and you have
∫ 1/(u^3+u) du
= ∫ 1/(u(u^2+1)) du
= ∫ 1/u - u/(u^2+1) du
= ln(u) - 1/2 ln(u^2+1)
= ln(e^x) - 1/2 ln(e^2x + 1)
= x - 1/2 ln(e^2x + 1)
Let u=e^x, du = e^x dx and you have
∫ 1/(u^3+u) du
= ∫ 1/(u(u^2+1)) du
= ∫ 1/u - u/(u^2+1) du
= ln(u) - 1/2 ln(u^2+1)
= ln(e^x) - 1/2 ln(e^2x + 1)
= x - 1/2 ln(e^2x + 1)