We can use the same substitution as above to calculate desired integral directly and get
integral( 0, 1, f(x)/√(1x^{2}) dx ) = integral( 0, π/2, f(sin(t))dt )
The right term can also be written as integral( 0, π/2, f(cos(t))dt ) (use cos = π/2sin)
Integrating
f(sin(t)) + f(cos(t)) = 2
from 0 to π/2 gives
integral( 0, π/2, f(sin(t)) + f(cos(t)) dt ) = 2*π/2
or
2*integral( 0, π/2, f(sin(t)) dt ) = π
So also in the general situation, the required integral value is π/2

Posted by JLo
on 20191214 07:20:11 