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

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

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integral of x*sqrt(x+1)

\int x \cdot x + 1 d x

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

Solution steps

\int x x + 1 d x

Apply u-substitution

: \int (u - 1) u d u

= \int (u - 1) u d u

Expand

(u - 1) u : u^{\frac{3}{2}} - u

= \int u^{\frac{3}{2}} - u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int u^{\frac{3}{2}} d u - \int u d u

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

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

= \frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = x + 1

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

Add a constant to the solution

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

\int \frac{1}{3 cos ( 5 x - \frac{π}{4} )} d x

\frac{d}{d x} ((x)^{l n (x)})

\int arctan (y) d y

f (x) = x^{4} + 5 e^{x}, (0, 5)

cos^{2} (x) sin (x) (\frac{d y}{d x}) + (cos^{3} (x)) y = 1

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