Cal II Integration: Solve (t+7)/(5-t)^(1/2)

[CallingAllCars7]

Homework Statement

find the following integral:

(t+7)/(5-t)^(1/2)

Homework Equations

maybe U*v'= uv-[ u'-v] or substitution

The Attempt at a Solution

Ive been trying substitution
u= (5-t)^(1/2)

t= 5-u^2
dt= -2u

((5-u^2)+7)/u )*-2u(du)

I get stuck here and I'm not even sure if I'm on the right track. I've seen the solution but I just can't get the answer.

 

[Mindscrape]

Have you tried integration by parts?

 

[Curious3141]

Rearrange it algebraically first and it becomes a doddle.

\frac{t+7}{\sqrt{5-t}} = -\frac{(5-t)-12}{\sqrt{5-t}} = -{{(5-t)}^{\frac{1}{2}} + 12{({5-t})^{-\frac{1}{2}}

 

[HallsofIvy]

Homework Statement

find the following integral:

(t+7)/(5-t)^(1/2)

Homework Equations

maybe U*v'= uv-[ u'-v] or substitution

The Attempt at a Solution

Ive been trying substitution
u= (5-t)^(1/2)

t= 5-u^2
dt= -2u

((5-u^2)+7)/u )*-2u(du)

I get stuck here and I'm not even sure if I'm on the right track. I've seen the solution but I just can't get the answer.

Why would you get stuck here? 5- u²+ 7= 12- u^2 and the u in the denominator cancels the u in "-2udu". Simple algebra gives
(-23+ 2u^2)du

 

[CallingAllCars7]

its just that the solution looks nothing like that :

its -2t(5-t)^(1/2)- (4/3)(5-t)^(3/2)-14(5-t)^(1/2)+c


I did also try integration by parts with another different answer.

 

[Gib Z]

Personally I would not have been clever enough to arrange it as Curious3141 did, but Thats the way I would like to have done it :)

I would have split it into 2 integrals, by splitting up the numerator. Then I would have tried t= 5 sin x, dt= 5 cos x dx on the first one, and u=5-t on the second. It gets you there, but Curious did it much better.