Do you have knowledge of complex analysis or only the basic (real) calculus?
Anyone know how to solve the following problems?
1. Integrate
∫(x-1) / (lnx) dx Limit 0≦x≦1
∫(x^Alpha)*lnx dx Limit 0≦x≦1
∫(sin x)^3 / (x^2) dx Limit 0≦x≦∞
∫((sin x )/ x) dx Limit 0≦x≦∞
2. (a) Evaluate ∫((sin x)^3) /((sin x)^3 + (cos x)^3) dx Limit 0≦x≦Pi/2
(b) What is the rate of change of f(x) = arctan (tan x) at x =Pi/2
3. A point mass m moves along the curve y = 2(X^2) due to gravity and no other external force is present. Determine the power of gravity as a function of Xo, g, t. (Xo means x knot)
Thanks!
Since is inverse function for we have,Originally Posted by thebigshow500
thus, the derivative (rate of change) of this is . The problem, is of course that is not in the domain of thus, it is not differenciable at thus, I would say that the rate of change does not exist there.
Use "Weierstrauss Substitution"Originally Posted by thebigshow500
Thus, for
Then,
This, converts any rational function of sine and cosine into an ordinary rational function.
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Let me continue,
first this function is countinous over thus, the integral is not improper of the second type. Thus, you may simply use the fundamental theorem of calculus.
Thus, the problem,
transfroms after Weierstrauss substitution into,
Multiply the top and bottom of the left fraction by to get,
Hope, this helps, it is still a mess.
You need,Originally Posted by thebigshow500
Express, the sine as an infinite series and divide by to get,
Upon, integration,
I just do not know what this infinite sum is equal to. It seems to be to diverge. Thus,
diverges.
HereOriginally Posted by TD!
http://www.padowan.dk/graph/
This is best program out there (freeware )
You're probably referring to some integrals which happened to be doable using complex analysis. Since calculus/analysis is one of my favourite fields in math, I replied there
It is indeed very powerful, many 'real integrals' which are very hard or even impossible using normal calculus can be computed with complex analysis. Like this one, since sin(x)/x doesn't have a primitive function (at least not in terms of the elementary functions).