math community,
I've been working on a few problems and have tried a substitution method that I thought would work but doesn't! Here is an example of it in action:
$\int \frac{x^2}{\sqrt{25 + x^2}} dx $
Substitute: $5u = x, 5du = dx$
$5 \int \frac{25u^2}{\sqrt{25+25u^2}}dx$
$=25 \int\frac{u^2}{\sqrt{1+u^2}}dx$
Substitute: $w = \sqrt{1+u^2}, dw = \frac{1}{\sqrt{1+u^2}}du$
$25\int{\frac{u^2}{\sqrt{u^2+1}}\sqrt{1+u^2}dw}$
$=25\int\frac{w^2-1}{w} w dw$ (where $u^2 = w^2-1$)
$= 25 \int (w^2 -1)dw $
$= 25\left(\frac{w^3}{3}-w\right)$
After substituting everything back in and simplifying, I get the following:
$-\frac{10}{3}\sqrt{25+x^2} + C$
Clearly,
$\frac{d}{dx}\left(-\frac{10}{3}\sqrt{25+x^2}\right)\neq\frac{x^2}{\sqrt{25 + x^2}}$
Where have I gone wrong in this method?
Thanks!