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

The integral of ((x^3)/2-x^{-1}+3sqrt(x)) is (x^4)/8-ln|x|+2x^{3/2}+C

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

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

\frac{x ^{4}}{8} - ln ∣ x ∣ + 2 x^{\frac{3}{2}} + C

Solution steps

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

\int \frac{x ^{3}}{2} d x = \frac{x ^{4}}{8}

\int x^{- 1} d x = ln ∣ x ∣

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

= \frac{x ^{4}}{8} - ln ∣ x ∣ + 2 x^{\frac{3}{2}}

Add a constant to the solution

= \frac{x ^{4}}{8} - ln ∣ x ∣ + 2 x^{\frac{3}{2}} + C

What is the integral of ((x^3)/2-x^{-1}+3sqrt(x)) ?
The integral of ((x^3)/2-x^{-1}+3sqrt(x)) is (x^4)/8-ln|x|+2x^{3/2}+C