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प्रश्न
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`.
पर्याय
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.
उत्तर
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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