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The cute substitution (Posted on 2019-12-13) Difficulty: 3 of 5
Given that f(x) satisfies the equation f(x)+f(√(1-x2))=2, compute the integral f(x)/√(1-x2) from 0 to 1.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Obvious substitution - 3 Comment 3 of 3 |
We can use the same substitution as above to calculate desired integral directly and get
integral( 0, 1, f(x)/√(1-x2) 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/2-sin)

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 2019-12-14 07:20:11
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