Question

deleted question.

Answer 1

Simple u-substitution :) but there is a little trick in the middle of it.

Answer 2

\[\Large\rm \color{royalblue}{u=1+2x}\]Subtracting 1 from each side,\[\Large\rm u-1=2x\]Divide by 2,\[\Large\rm \color{orangered}{\frac{1}{2}(u-1)=x}\]

Do you understand WHY I did that?
The blue let's us take care of the denominator.
The orange let's us take care of the numerator.

Answer 3

And you will also need your dx I suppose.\[\Large\rm du=2dx\qquad\to\qquad \frac{1}{2}du=dx\]

Answer 4

The orange :o

Answer 5

Woops, an extra 1/2 from your du as well, yes?

Answer 6

Yah there we go \c:/
The whole trick was, we needed to get rid of the addition under the root, so we could divide it easier.

Answer 7

\[\LARGE \int\limits\limits_{ }^{ }\frac{\frac{1}{2}(u-1)}{\sqrt{u}}~\left(\frac{1}{2}du\right)\]