Question

Solve this ! Topic: limits​

Answer 1

$\large\underline{\sf{Solution-}}$

Given integral is

$\red{\rm :\longmapsto\:\displaystyle\int_0^a\rm \{f(x) + f( - x) \}dx \: = \: \displaystyle\int_{ - a}^a\rm g(x) \: dx}$

Let we consider,

$\red{\rm :\longmapsto\:\displaystyle\int_0^a\rm \{f(x) + f( - x) \}dx \: }$

can be rewritten as

$\rm \: = \: \: \displaystyle\int_0^a\rm f(x) \: dx \: + \: \displaystyle\int_0^a\rm f( - x) \: dx$

In second integral, we use method of Substitution,

$\red{\rm :\longmapsto\:Let \: x = - y}$

$\red{\rm :\longmapsto\:dx = - dy}$

In definite integrals, when we substitute, the limits have to be changed accordingly.

So, limits become

$\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 0 & \sf 0 \\ \\ \sf a & \sf - a \end{array}} \\ \end{gathered}$

Now, above integral can be rewritten as

$\rm \: = \: \: \displaystyle\int_0^a\rm f(x) \: dx \: - \: \displaystyle\int_0^{ - a}\rm f(y) \: dy$

We know the Limit interchanging property of integrals,

$\boxed{ \rm{ \displaystyle\int_b^a\rm f(x) \: dx \: = \: - \: \displaystyle\int_a^b\rm f(x) \: dx}}$

So, above integral can be rewritten as

$\rm \: = \: \: \displaystyle\int_0^a\rm f(x) \: dx \: + \: \displaystyle\int_{ - a}^{0}\rm f(y) \: dy$

We know the variable changing property of integral,

$\boxed{ \rm{ \displaystyle\int_b^a\rm f(x) \: dx \: = \: \displaystyle\int_b^a\rm f(y) \: dy \: }}$

So, using this, the above integral can be rewritten as

$\rm \: = \: \: \displaystyle\int_0^a\rm f(x) \: dx \: + \: \displaystyle\int_{ - a}^{0}\rm f(x) \: dx$

can be further rewritten as

$\rm \: = \: \: \: \displaystyle\int_{ - a}^{a}\rm f(x) \: dx$

Hence, we concluded that

$\red{\rm :\longmapsto\:\displaystyle\int_0^a\rm \{f(x) + f( - x) \}dx \: = \: \displaystyle\int_{ - a}^a\rm f(x) \: dx}$

But given that,

$\red{\rm :\longmapsto\:\displaystyle\int_0^a\rm \{f(x) + f( - x) \}dx \: = \: \displaystyle\int_{ - a}^a\rm g(x) \: dx}$

So, from these we concluded that,

$\boxed{ \bf \: { f(x) \: = \: g(x) \: }}$

• Hence, Option (a) is correct