What is the integral of 2e^{-2x}+4x+1 ?

The integral of 2e^{-2x}+4x+1 is -e^{-2x}+2x^2+x+C

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integral of 2e^{-2x}+4x+1

\int 2 e^{- 2 x} + 4 x + 1 d x

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

Solution steps

\int 2 e^{- 2 x} + 4 x + 1 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 2 e^{- 2 x} d x + \int 4 x d x + \int 1 d x

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

\int 4 x d x = 2 x^{2}

\int 1 d x = x

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

Add a constant to the solution

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

What is the integral of 2e^{-2x}+4x+1 ?
The integral of 2e^{-2x}+4x+1 is -e^{-2x}+2x^2+x+C