What is the integral of 2xe^y+(2x)/y+1/(y^3) ?

The integral of 2xe^y+(2x)/y+1/(y^3) is x/(y^3)+(x^2)/y+e^yx^2+C

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integral of 2xe^y+(2x)/y+1/(y^3)

\int 2 x e^{y} + \frac{2 x}{y} + \frac{1}{y ^{3}} d x

\frac{x}{y ^{3}} + \frac{x ^{2}}{y} + e^{y} x^{2} + C

Solution steps

\int 2 x e^{y} + \frac{2 x}{y} + \frac{1}{y ^{3}} d x

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

\int \frac{1}{y ^{3}} d x = \frac{x}{y ^{3}}

\int \frac{2 x}{y} d x = \frac{x ^{2}}{y}

\int 2 e^{y} x d x = e^{y} x^{2}

= \frac{x}{y ^{3}} + \frac{x ^{2}}{y} + e^{y} x^{2}

Add a constant to the solution

= \frac{x}{y ^{3}} + \frac{x ^{2}}{y} + e^{y} x^{2} + C

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What is the integral of 2xe^y+(2x)/y+1/(y^3) ?
The integral of 2xe^y+(2x)/y+1/(y^3) is x/(y^3)+(x^2)/y+e^yx^2+C