What is the integral of (e^x)/(x^2)*e^x+2x ?

The integral of (e^x)/(x^2)*e^x+2x is -(e^{2x})/x+2\Ei(2x)+x^2+C

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integral of (e^x)/(x^2)*e^x+2x

\int \frac{e ^{x}}{x ^{2}} \cdot e^{x} + 2 x d x

- \frac{e ^{2 x}}{x} + 2 Ei (2 x) + x^{2} + C

Solution steps

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

\int \frac{e ^{x}}{x ^{2}} e^{x} d x = - \frac{e ^{2 x}}{x} + 2 Ei (2 x)

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

= - \frac{e ^{2 x}}{x} + 2 Ei (2 x) + x^{2}

Add a constant to the solution

= - \frac{e ^{2 x}}{x} + 2 Ei (2 x) + x^{2} + C

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What is the integral of (e^x)/(x^2)*e^x+2x ?
The integral of (e^x)/(x^2)*e^x+2x is -(e^{2x})/x+2\Ei(2x)+x^2+C