Question

\[\int_0^\pi {xf(\sin x)} dx = \] [IIT\[1982\]; Kurukshetra CEE\[1993\]]
E) \[\pi \int_0^\pi {f(\sin x)} dx\]
F) \[\dfrac{\pi }{2}\int_0^\pi {f(\sin x)} dx\]
G) \[\dfrac{\pi }{2}\int_0^{\dfrac{\pi }{2}} {f(\sin x)} dx\]
H) None of these

Answer 1

Hint: in this question, we have to find the given integral. In order to find this, the properties of the definite integral are used. From appropriate property of definite integral given integration is evaluated.

Formula Used: The definite integral is the area under the curve between two fixed limits in which one limit is upper limit and other limit is lower limit.
Property of definite integral used is given as
\[\int_0^a {xf(x)} dx = \dfrac{1}{2}a\int_0^a {f(x)} dx\]
If given function satisfy the condition\[f(a - x) = f(x)\]then we will use the above property of definite integral
Where
a is upper limit of integral and 0is a lower limit of integral.

Complete step by step solution: Given: Definite integral \[\int_0^\pi {xf(\sin x)} dx\]
Here in this integral upper limit is \[\pi \] and lower limit of integral is zero
Now check the condition \[f(a - x) = f(x)\]
\[\sin (\pi - x) = \sin (x)\]
Condition is satisfied
We know that
\[\int_0^a {xf(x)} dx = \dfrac{1}{2}a\int_0^a {f(x)} dx\] If \[f(a - x) = f(x)\]
\[\int_0^\pi {xf(\sin x)} dx = \dfrac{\pi }{2}\int_0^\pi {f(\sin x)} dx\]
So required integral is
\[\dfrac{\pi }{2}\int_0^\pi {f(\sin x)} dx\]

Option ‘B’ is correct

Note: Here we must check that the given functions satisfy the condition \[f(a - x) = f(x)\]or not if functions satisfy the condition then only we apply the property.
The definite integral is the area under the curve between two fixed limits.
Let f(x) is a function and suppose integration of function f(x) is F(x) then definite integral of f(x) having upper limit b and lower limit a can be written in mathematical expression as
\[\int_a^b {f(x)} dx = F(b) - F(a)\]
Properties of the definite integrals are:
1) Interchanging the upper and lower limit: \[\int_b^a {f(x)} dx = - \int_a^b {f(x)} dx\]
2) \[\int_b^a {f(x)} dx = \int_b^a {f(t)} dt\]
3) \[\int_0^a {f(x)} dx = \int_0^a {f(a - x)} dx\]
4) \[\int_a^b {f(x)} dx = \int_a^c {f(x)} dx + \int_c^b {f(x)} dx\]