evaluate the indefinite integral with subsitution

Dec 2009
103
0
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
Feb 2009
2,763
1,146
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.
 
May 2010
20
8
You chose the wrong u


\(\displaystyle u=2x+x^2\)

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

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

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

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