What is the integral of (y^2-y^3e^y+sqrt(y))/(y^3) ?

The integral of (y^2-y^3e^y+sqrt(y))/(y^3) is ln|y|-e^y-2/(3y^{3/2)}+C

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integral of (y^2-y^3e^y+sqrt(y))/(y^3)

\int \frac{y ^{2} - y ^{3} e ^{y} + y}{y ^{3}} d y

ln ∣ y ∣ - e^{y} - \frac{2}{3 y ^{\frac{3}{2}}} + C

Solution steps

\int \frac{y ^{2} - y ^{3} e ^{y} + y}{y ^{3}} d y

Expand

\frac{y ^{2} - y ^{3} e ^{y} + y}{y ^{3}} : \frac{1}{y} - e^{y} + \frac{1}{y ^{\frac{5}{2}}}

= \int \frac{1}{y} - e^{y} + \frac{1}{y ^{\frac{5}{2}}} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{y} d y - \int e^{y} d y + \int \frac{1}{y ^{\frac{5}{2}}} d y

\int \frac{1}{y} d y = ln ∣ y ∣

\int e^{y} d y = e^{y}

\int \frac{1}{y ^{\frac{5}{2}}} d y = - \frac{2}{3 y ^{\frac{3}{2}}}

= ln ∣ y ∣ - e^{y} - \frac{2}{3 y ^{\frac{3}{2}}}

Add a constant to the solution

= ln ∣ y ∣ - e^{y} - \frac{2}{3 y ^{\frac{3}{2}}} + C

What is the integral of (y^2-y^3e^y+sqrt(y))/(y^3) ?
The integral of (y^2-y^3e^y+sqrt(y))/(y^3) is ln|y|-e^y-2/(3y^{3/2)}+C