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

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

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integral of e^{-4x}+x+(4x+1)

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

- \frac{1}{4} e^{- 4 x} + \frac{x ^{2}}{2} + 2 x^{2} + x + C

Solution steps

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

\int e^{- 4 x} d x = - \frac{1}{4} e^{- 4 x}

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

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

\int 1 d x = x

= - \frac{1}{4} e^{- 4 x} + \frac{x ^{2}}{2} + 2 x^{2} + x

Add a constant to the solution

= - \frac{1}{4} e^{- 4 x} + \frac{x ^{2}}{2} + 2 x^{2} + x + C

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