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

The integral of x(x+1)^{1/2} is 2/3 x(x+1)^{3/2}-4/15 (x+1)^{5/2}+C

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

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

\frac{2}{3} x (x + 1)^{\frac{3}{2}} - \frac{4}{15} (x + 1)^{\frac{5}{2}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{2}{3} (x + 1)^{\frac{3}{2}} d x = \frac{4}{15} (x + 1)^{\frac{5}{2}}

= \frac{2}{3} x (x + 1)^{\frac{3}{2}} - \frac{4}{15} (x + 1)^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{3} x (x + 1)^{\frac{3}{2}} - \frac{4}{15} (x + 1)^{\frac{5}{2}} + C

\frac{\partial}{\partial x} (y^{0.75} x^{0.25})

f (x) = 9 e^{5} x + e^{- x^{2}}, at x = 1

\frac{d y}{d x} = 6 x^{2} y^{2}

x \to 2 - lim (4 - x^{2})

f (r) = \frac{5}{r ^{3}}

What is the integral of x(x+1)^{1/2} ?
The integral of x(x+1)^{1/2} is 2/3 x(x+1)^{3/2}-4/15 (x+1)^{5/2}+C