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

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

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

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

2 ln ∣ x ∣ + \frac{1}{2} arctan (2 x) + \frac{2}{x ^{\frac{1}{2}}} + C

Solution steps

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

\int \frac{1}{4 x ^{2} + 1} d x = \frac{1}{2} arctan (2 x)

\int x^{- \frac{3}{2}} d x = - \frac{2}{x ^{\frac{1}{2}}}

= 2 ln ∣ x ∣ + \frac{1}{2} arctan (2 x) - (- \frac{2}{x ^{\frac{1}{2}}})

Simplify

= 2 ln ∣ x ∣ + \frac{1}{2} arctan (2 x) + \frac{2}{x ^{\frac{1}{2}}}

Add a constant to the solution

= 2 ln ∣ x ∣ + \frac{1}{2} arctan (2 x) + \frac{2}{x ^{\frac{1}{2}}} + C

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