Question

Help with integrals please Question attached

Answer 1

the description in the question is actually very good, use substitution, make sure that if you make a substitution then you differentiate also:
\[\Large u=2-x \implies x=2-u \implies \frac{dx}{du}=-1 \implies dx=-du\] now substitute all these expressions back into your integral:
\[ \Large \int ((2-u+1)\sqrt{u} (-du)\] simplify and then try to solve it.

Answer 2

ohh ok

Answer 3

hope that helps, if you simplify it and distribute it out you will end up with integrals of the form \(\int u^ndu , \ n \in \mathbb{Q}\) which can easily be solved using basic integration methods.

Answer 4

Another substitution leads to ...
$$
u = \sqrt{2-x}
$$
then
$$
du = -\cfrac{1}{2\sqrt{2-x} } dx\\
$$
so
$$
\int (x+1) \sqrt{2-x} \, dx=
-2 \int u^2 (3-u^2) du
$$
Which might be easier to integrate.

Answer 5

Ok thanks

Answer 6

another very clean and elegant approach to this problem (+1). You might want to try out both substitutions @Rea201, with integration it's all about developing an eye for the various integral types and the proper substitutions. The ones you feel most comfortable with, or feel the most intuitive to you are usually the ones to go with :-)

Answer 7

ok thanks for the advice