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

The integral of (x^{3/2}-2x^3+sqrt(x)) is 2/5 x^{5/2}-(x^4)/2+2/3 x^{3/2}+C

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

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

\frac{2}{5} x^{\frac{5}{2}} - \frac{x ^{4}}{2} + \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

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

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

= \frac{2}{5} x^{\frac{5}{2}} - \frac{x ^{4}}{2} + \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{5} x^{\frac{5}{2}} - \frac{x ^{4}}{2} + \frac{2}{3} x^{\frac{3}{2}} + C

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What is the integral of (x^{3/2}-2x^3+sqrt(x)) ?
The integral of (x^{3/2}-2x^3+sqrt(x)) is 2/5 x^{5/2}-(x^4)/2+2/3 x^{3/2}+C