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 apply integration by parts to the integrand (integrate x and differentiaete the other factor) simplify (changing occurrence of sin^2(x) to 1-cos^2(x)) will give the formula in part .

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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