This 10part problem will most certainly be an exhilarating experience for the mathematically inclined, but could also possibly be the stuff of nightmares for the average student of Calculus. Note that a small implementation of reduction formula is present.
Screenshot
http://www.alevelmaths.com/extremeproblems.htm
(see Q 11, solutions provided in pdf format)
Have fun. Peace.
Extreme problem on Integration Techniques
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whitecorp  BrownBelt
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Re: Extreme problem on Integration Techniques
by JWMorgan » Tue May 24, 2016 9:26 pm
Not an extreme problem. The normal stuff in Pure Math papers in the 60's. Part (a): tear out cos^2(x) from the integrand write in terms of cos(2x), expand and apply integration by parts and simplify. Part (b) integration by parts on the integrand will yield the formula directly. For part (c) apply integration by parts to the integrand (integrate x and differentiaete the other factor) simplify (changing occurrence of sin^2(x) to 1cos^2(x)) will give the formula in part (c).
It is a good practice in integration by parts, in each part, there is only one integration by parts to perform. Parts (d), (e) and (f) straightforward. Part (g) requires an integration by parts and putting the parts together gives part(i) and the Squeeze theorem finishes the last part. This is Euler's formula.
It is a good practice in integration by parts, in each part, there is only one integration by parts to perform. Parts (d), (e) and (f) straightforward. Part (g) requires an integration by parts and putting the parts together gives part(i) and the Squeeze theorem finishes the last part. This is Euler's formula.

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