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

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

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integral of x(sqrt(x)+1)

\int x (x + 1) d x

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

Solution steps

\int x (x + 1) d x

Expand

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

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

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

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

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

Add a constant to the solution

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

\frac{\partial}{\partial y} (ln (x))

n = 1 \sum \infty 4^{n}

y = 4 x^{2}, (2, 16)

\int_{1}^{10} \frac{1}{x - 6} d x

f (x) = 5 x^{2}, at x = - 3, - 45

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