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

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

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

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

2 e^{x^{2}} x + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Simplify

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

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

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

\int 2 e^{x^{2}} x^{2} d x = e^{x^{2}} x - \frac{π}{2} erfi (x)

\int e^{x^{2}} d x = \frac{π}{2} erfi (x)

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

Add similar elements:

- \frac{π}{2} erfi (x) + \frac{π}{2} erfi (x) = 0

= 2 e^{x^{2}} x

Add a constant to the solution

= 2 e^{x^{2}} x + C

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