I did this problem in my STEP exam this summer. It's really good fun and I'm sure you'll enjoy it.
Given that;
sec2(41π−21x)=1+sin(x)2
and
∫0πxf(sinx)dx=2π∫0πf(sinx)dx
Evaluate ∫0π1+sinxxdx
and ∫0π(1+sinx)22x3−3πx2dx
What can I deduce about f(sin(x))? If the original function was f(x)=1 for example, then f(sin(x)) has no effect. Obviously, the original function wasn't so, but what can be deduced?
I did it by a substitution; there may be a quicker way. Express 2x^2 + 4x + 3 in terms of (x + 1)^2 and the kind of substitution to do might jump out at you.
I did it by a substitution; there may be a quicker way. Express 2x^2 + 4x + 3 in terms of (x + 1)^2 and the kind of substitution to do might jump out at you.
I did it by a substitution; there may be a quicker way. Express 2x^2 + 4x + 3 in terms of (x + 1)^2 and the kind of substitution to do might jump out at you.
Thanks. For 10, what's the suitable, non-trigonometric substitution? I tried integration by parts, but 20 minutes later and I'm nowhere.