What is the integral of (-4x^3+2e^{-2x}+1/(2x)) ?

The integral of (-4x^3+2e^{-2x}+1/(2x)) is -x^4-e^{-2x}+1/2 ln|x|+C

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integral of (-4x^3+2e^{-2x}+1/(2x))

\int (- 4 x^{3} + 2 e^{- 2 x} + \frac{1}{2 x}) d x

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

Solution steps

\int - 4 x^{3} + 2 e^{- 2 x} + \frac{1}{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 4 x^{3} d x + \int 2 e^{- 2 x} d x + \int \frac{1}{2 x} d x

\int 4 x^{3} d x = x^{4}

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

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

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

Add a constant to the solution

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

\frac{d y}{d x} + 4 (tan (4 x)) y = 10 cos (4 x)

x \to 0 lim (\frac{x}{( 4 ^{x} - 2 ^{x} )})

\frac{d}{d x} (arcsin (2 x^{5}))

x^{2} + 9

\frac{d}{d x} ((5 x^{3} + 5) (x^{4} - 3 x))

What is the integral of (-4x^3+2e^{-2x}+1/(2x)) ?
The integral of (-4x^3+2e^{-2x}+1/(2x)) is -x^4-e^{-2x}+1/2 ln|x|+C