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

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

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

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

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

Solution steps

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

Expand

x (x^{\frac{3}{2}} + x) : x^{2} + x^{\frac{3}{2}}

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

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

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

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

Add a constant to the solution

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

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