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`.

विकल्प

• 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.

Answer 1

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)`