anonymous
  • anonymous
Evaluate the intragural. 3y(7+2y^2)^1/2 dy
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Does anyone have any idea?
TuringTest
  • TuringTest
\[u=7+2y^2\]\[du=4ydy\]\[\int3y(7+2y^2)^{1/2}dy=\frac34\int u^{1/2}du\]
anonymous
  • anonymous
Wait. So how did that 3y in front of the u just dissapear?

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TuringTest
  • TuringTest
I just pulled it outside the integral sign
TuringTest
  • TuringTest
\[\int cf(x)dx=c\int f(x)dx\]
anonymous
  • anonymous
Even though its 3y(u)^1/2? I thought if it was 3+ you could
anonymous
  • anonymous
Im just not getting where the y went
TuringTest
  • TuringTest
\[u=7+2y^2\]\[du=4ydy\to ydy=\frac14du\]\[\int3y(7+2y^2)^{1/2}dy=3\int (7+2y^2)^{1/2}ydy\]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.
TuringTest
  • TuringTest
a little slower on the last step:\[3\int (7+2y^2)^{1/2}ydy=3\int(u)^{1/2}(\frac14du)=\frac34\int u^{1/2}du\]
anonymous
  • anonymous
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?
TuringTest
  • TuringTest
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
TuringTest
  • TuringTest
\[3\int (7+2y^2)^{1/2}(ydy)=3\int(u)^{1/2}(\frac14du)=\frac34\int u^{1/2}du\]see how ydy got replaced?
anonymous
  • anonymous
Oh I think I get it. Thanks a lot
TuringTest
  • TuringTest
u-subs can be tricky at first, but with practice it's a flash :D

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