# evaluate the indefinite integral with subsitution

#### dorkymichelle

evaluate the indefinite integral with subsitution
$$\displaystyle 11.\int(x+1)\sqrt{2x+x^2}dx$$
$$\displaystyle u=x+1$$
$$\displaystyle du=1dx$$
$$\displaystyle 2x+x^2=u^2-1$$
$$\displaystyle u\sqrt{u^2-1}du$$
$$\displaystyle u(u^2-1)^1/2du$$
$$\displaystyle 1/2u^2*2/3(u^2-1)^3/2$$
$$\displaystyle 1/3u^2(u^2-1)^3/2$$
$$\displaystyle 1/3(2x+x^2+1)(2x+x^2)^3/2$$
but the answer is simply $$\displaystyle 1/3(2x+x^2)^3/2$$
what did i do wrong

#### matheagle

MHF Hall of Honor
You seem to have integrated $$\displaystyle \int f(x)g(x)dx$$

as $$\displaystyle \int f(x)dx\int g(x)dx$$ which is wrong.

But leaving out the integral signs makes this very hard to read.

Try completing the square inside the square root and then letting u=x+1.

#### DrDank

You chose the wrong u

$$\displaystyle u=2x+x^2$$

$$\displaystyle du=2(x+1)dx$$

$$\displaystyle \int(x+1)\sqrt{2x+x^2}dx$$

$$\displaystyle \int\frac{(x+1)\sqrt{u}}{2(x+1)} du$$

$$\displaystyle \frac{1}{2}\int\sqrt{u} du$$

HallsofIvy