1. $\displaystyle \int xdx / \sqrt{4+5x} $.
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Well there're lots of ways of doing this. I keep the following: put $\displaystyle t=\sqrt{4+5x}.$
I tried plugin u=4+5 and dv=x but I got a mess. Can someone show me?
But you're integrating by parts, that's not necessary in this case. We can write my substitution as $\displaystyle x=\frac{t^2-4}5,$ now $\displaystyle dx=\frac{2t}5\,dt.$ Substitute those in the integral now, what do you get?
$\displaystyle \int \frac{x}{\sqrt{4+5x}}~dx$ $\displaystyle t=\sqrt{4+5x} \implies x = \frac{t^2-4}{5}$ Now $\displaystyle \int \frac{\frac{t^2-4}{5}}{t}~dx=\int \frac{t^2-4}{5t}\frac{2t}{5}~dt$
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