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Question
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Options
Both (A) and (R) are true and (R) is the correct explanation of (A).
Both (A) and (R) are true, but (R) is not the correct explanation of (A).
(A) is true, but (R) is false.
(A) is false, but (R) is true.
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Solution
Both (A) and (R) are true and (R) is the correct explanation of (A).
Explanation:
I = `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` ...(i)
Using property of definite integral
`int_a^b f(x) dx = int_a^b f(a + b - x) dx`
I = `int_2^8 sqrt(x)/(sqrt(10 - x) + sqrt(x))dx` ...(ii)
Adding equations (i) and (ii)
2I = `int_2^8 (sqrt(10 - x) + sqrt(x))/(sqrt(10 - x) + sqrt(x))dx`
= `int_2^8 dx`
= `[x]_2^8`
= 8 – 2
= 6
`\implies` I = 3
R is also true as the property P4 is
`int_a^b f(x)dx = int_a^b f(a + b - x)`
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