What is the integral of xsqrt(x+3) ?

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

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integral of xsqrt(x+3)

\int x x + 3 d x

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

Solution steps

\int x x + 3 d x

Apply u-substitution

: \int (u - 3) u d u

= \int (u - 3) u d u

Expand

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

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

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

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

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

Substitute back

u = x + 3

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

Add a constant to the solution

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

\frac{d y}{d x} = \frac{( x + 3 y )}{y - 3 x}

x \to 3 lim (\frac{x ^{3} - 6 x ^{2} + 9}{x ^{2} - 9})

y = 7 x - 4 x^{2}, (- 3, - 57)

x \to 999 lim (x^{2} - 1)

\int \frac{4 x}{1 - 2 x} d x

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