What is the integral of e^{ln(x)}*e^{2x}(x^{-1}+2) ?

The integral of e^{ln(x)}*e^{2x}(x^{-1}+2) is e^{2x}x+C

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integral of e^{ln(x)}*e^{2x}(x^{-1}+2)

\int e^{l n (x)} \cdot e^{2 x} (x^{- 1} + 2) d x

e^{2 x} x + C

Solution steps

\int e^{l n (x)} e^{2 x} (x^{- 1} + 2) d x

Simplify

e^{l n (x)} e^{2 x} (x^{- 1} + 2) : e^{2 x} + 2 e^{2 x} x

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

\int e^{2 x} d x = \frac{1}{2} e^{2 x}

\int 2 e^{2 x} x d x = e^{2 x} x - \frac{1}{2} e^{2 x}

= \frac{1}{2} e^{2 x} + e^{2 x} x - \frac{1}{2} e^{2 x}

Simplify

\frac{1}{2} e^{2 x} + e^{2 x} x - \frac{1}{2} e^{2 x} : e^{2 x} x

= e^{2 x} x

Add a constant to the solution

= e^{2 x} x + C

What is the integral of e^{ln(x)}*e^{2x}(x^{-1}+2) ?
The integral of e^{ln(x)}*e^{2x}(x^{-1}+2) is e^{2x}x+C