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प्रश्न
Evaluate the following using properties of definite integral:
`int_0^(i/2) (sin^7x)/(sin^7x + cos^7x) "d"x`
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उत्तर
Using the property
`int_0^"a" "f"(x) "d"x = int_0^"a" "f"("a" - x) "d"x`
Let I = `int_0^(pi/2) (sin^7x)/(sin^7x + cos^7x) "d"x` ........(1)
I = `int_0^(pi/2) (sin^7(pi/2 - x))/(sin^7(pi/2 - x) + cos^7(pi/2 - x)) "d"x`
I = `int_0^(pi/2) (cos^7x)/(cos^7x + sin^x) "d"x` .........(2)
Adding (1) and (2)
I + I = `int_0^(pi/2) (sin^7x + cos^7x)/(sin^7x + cos^7x "d"x`
2I `int_0^(pi/2) "d"x`
2I = `[x]_0^(pi/2) = [pi/2 - 0]`
2I = `pi/2`
⇒ I = `pi/4`
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