What is the integral of x(2sqrt(x)+1) ?

The integral of x(2sqrt(x)+1) is 4/5 x^{5/2}+(x^2)/2+C

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integral of x(2sqrt(x)+1)

\int x (2 x + 1) d x

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

Solution steps

\int x (2 x + 1) d x

Expand

x (2 x + 1) : 2 x^{\frac{3}{2}} + x

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

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

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

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

Add a constant to the solution

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

n = 1 \sum \infty \frac{1}{2 n + 4}

\int \frac{1}{sin ( x ) sin ( 2 x )} d x

\int 2 x (x^{2} + 2)^{- 5} d x

x^{5} y^{^{'}} + y^{5} = 0

x \to - 1 lim (e^{0})

What is the integral of x(2sqrt(x)+1) ?
The integral of x(2sqrt(x)+1) is 4/5 x^{5/2}+(x^2)/2+C