Question

Evaluate the intragural. 3y(7+2y^2)^1/2 dy

Answer 1

Does anyone have any idea?

Answer 2

\[u=7+2y^2\]\[du=4ydy\]\[\int3y(7+2y^2)^{1/2}dy=\frac34\int u^{1/2}du\]

Answer 3

Wait. So how did that 3y in front of the u just dissapear?

Answer 4

I just pulled it outside the integral sign

Answer 5

\[\int cf(x)dx=c\int f(x)dx\]

Answer 6

Even though its 3y(u)^1/2? I thought if it was 3+ you could

Answer 7

Im just not getting where the y went

Answer 8

\[u=7+2y^2\]\[du=4ydy\to ydy=\frac14du\]\[\int3y(7+2y^2)^{1/2}dy=3\int (7+2y^2)^{1/2}ydy\]I just pulled the three out and put ydy together, so now I can sub in my expressions for 7+2y^2 and ydy that I got above:\[3\int (7+2y^2)^{1/2}ydy=\frac34\int u^{1/2}du\]let me know if something is still eluding you.

Answer 9

a little slower on the last step:\[3\int (7+2y^2)^{1/2}ydy=3\int(u)^{1/2}(\frac14du)=\frac34\int u^{1/2}du\]

Answer 10

I sorta get it I think. Just not sure how ydy becames (1/4du) when du did equal 4ydy. Shouldnt it be (1/4du)y?

Answer 11

the u sub works such that we need to wind up with \[\int u^ndu\]watch the terms that we get from our substitution\[u=7+2y^2\]\[du=4ydy\]\[ydy=\frac14du\]so we rearrange our integral so that we replace ydy by 1/4du
if we had ydu we couldn't integrate, so that would not help

Answer 12

\[3\int (7+2y^2)^{1/2}(ydy)=3\int(u)^{1/2}(\frac14du)=\frac34\int u^{1/2}du\]see how ydy got replaced?

Answer 13

Oh I think I get it. Thanks a lot

Answer 14

u-subs can be tricky at first, but with practice it's a flash :D