should i use substitution?? if yes how should should i use it? plz i need some directions?
k plz someone?...so far i used trig. substitution. i got a=8, so i used x=asin(è)so according to this substitution i got x=8sin(è) and dx=8cos(è) dè...then i used used the plug them in the original problem ...
so when i did that and simplfying it...i got 128sin^2(è)*8cos(è)/squarroot of 64-256sin^2...then how do i intergrate that? i am doing my best... am i doing it correct?
Are you certain the x in the numerator is squared? If it is not, then this helps...
divide numerator and denominator by 1/2
then you have INT x^2/sqrt(16-x^2)
Now, integrate it by parts, u= x^2, v= 1/sqrt(16-x^2) You will get power function, and an arcsin function.
integrate (2x^2)/squarroot of (64-4x^2)
no IT IS squared...shouldnt i be using trignometric substitution?
Yes, use substitution.
Let u = x^2
du = 2 x dx
x = sqrt u
dx = (1/2)/u^(1/2) du
and you get
(Integral of)[ 2u/sqrt(64 - 2u)](1/2)/u^(1/2) du
= (Integral of) sqrt(u/(64 - 2u) du
Letting v = 64 - 2u will simplify that further.
If it is not squared, yes, use trig substitution.
so integration by parts would help? or can i still use trignometric substittion?
i did use it...i got to the point of 128sin^2*8cos(è) d(è)/6-16sin(è) ...how do i integrate this?
why would v=64-2u? becasue it is square root of 64-4x^2...so wouldnt v=8-2x? i am confused on that part...but plz so far i used trig. substitution...i got to the point 128sin^2è*8cosè dè /8-16sinè ? but i am not sure where to go after that in order to integrate it?...
anyone?
i got the answer to be 2sin^2(è)....is that correct? i used trig. substitution
square root of1/2
To integrate the expression you have, you started with trigonometric substitution by letting x = 8sin(è) and dx = 8cos(è) dè. This substitution simplified the integral, and you obtained the expression 128sin^2(è)*8cos(è) / squareroot of 64-256sin^2(è).
To integrate this expression further, you can simplify the numerator by factoring out 64 from the squareroot, which gives you 128sin^2(è)*8cos(è) / sqrt(64(1 - 4sin^2(è))). Simplifying this further, you have 1024sin^2(è)cos(è) / (8sqrt(1 - 4sin^2(è))).
Now, to integrate this expression, you can use the trigonometric identity sin^2(è) = (1 - cos(2è)) / 2. By substituting this in, the integral becomes 512cos(è)(1 - cos(2è)) / sqrt(1 - 4sin^2(è)).
At this point, the integral is more complicated and may require further manipulation or the use of other integration techniques, such as integration by parts. You can use the trigonometric identity cos(2è) = 1 - 2sin^2(è) to simplify the expression further and potentially make the integration process easier.
Remember to always check your answer by differentiating it to see if you get the original expression back.