Question

Integral sin2x*(1+4cos^2 x)^1/2 dx

Answer 1

Hint : there is an obvious substitution here

Answer 2

u = 1+ 4cos^2 x ?

Answer 3

that should work!

Answer 4

But when I take the derivitive of that for U substitution I get 8cosxsinx which I am unsure if that helps me...

Answer 5

recall the trig identity
\[ \sin (2x) = 2\sin x \cos x\]

Answer 6

Thanks!!!!

Answer 7

\[u = 1+4\cos^2x \\~\\\implies du = 4(2\cos x (-\sin x)) dx = -8 \sin x \cos xdx = -4\sin (2x) dx\\~\\\implies \sin(2x) dx = -\frac{du}{4}\]

Answer 8

the integral becomes
\[\int \sqrt{u} (-\dfrac{du}{4}) = -\frac{1}{4}\int \sqrt{u}~du\]

Answer 9

-1/6 U^3/2

Answer 10

looks good! dont forget to substitute back the \(u\)

Answer 11

May be a little late to ask (; Did I set up the right integral for what this problem was asking?? @ganeshie8

Answer 12

Also evaluating my integral gives me zero and wolphram has an odd form on the answer...

Answer 13

somethings not right. you obviously shouldn't get 0 as the mass of the wire?

Answer 14

I am wondering if it is my initial integral set up...

Answer 15

Or am I just makking a mistake towards the end trying to evaluate it?

Answer 16

your setup looks good, i see two mistakes in the integrand however..

Answer 17

I got it ds should be \[\sqrt{1+4\cos ^{2}2t}\]

Answer 18

1) you forgot about absolute valuebars around the density : |sin(2x)|
2) see alekos reply above

Answer 19

so the integral changes somewhat

Answer 20

would it still be u sub at that point?

Answer 21

don't think that's going to work. let me think

Answer 22

wolfram provides an answer of 5.91577

Answer 23

Is this the answer that you would expect?

Answer 24

Makes more sense than zero but not sure how to calculate integral without wolphrum.

Answer 25

yes. that is the question. still working on it

Answer 26

Thanks Alekos. I gotta get some sleep.

Answer 27

what time is it over there?

Answer 28

so according to wolfram we have
\[-1/4(\sqrt{2\cos4t+3 })\cos2t - 1/8Ln(2\cos2t + \sqrt{2\cos4t+3})\]

Answer 29

step by step solution attached.
in order to get the final result integrate over (0,pi/2) and multiply by 4