QJEEM19Ma3d95: JEE Main 2019 Mathematics - Class 12 JEEM19Ma3d95 Solution: Let where g is a non-zero even function. If ƒ(x + 5) = g(x),…

Question

Verified Solution • 2025–26 Session moderate

Let

f(x) = \int\limits_0^x {g(t)dt}

where g is a non-zero even function. If ƒ(x + 5) = g(x), then

\int\limits_0^x {f(t)dt}

equals-

Accepted Answer

f(x) = \int\limits_0^x {g(t)dt}

f\left( { - x} \right) = \int\limits_0^{ - x} {g\left( t \right)} dt

Put t = -v

=

- \int\limits_0^x {g\left( { - v} \right)} dv

=

- \int\limits_0^x {g\left( v \right)} d(v)

[ as g(v) is an even function.]

= - f(x)

\Rightarrow

f(-x) = -f(x)

\therefore

f(x) is an odd function.

Given ƒ(x + 5) = g(x)

\therefore

g(- x) = ƒ(- x + 5)

\Rightarrow

g(x) = - f(x - 5) [as g(x) is even and f(x) is an odd function]

Replacing x by x + 5, we get

f(x) = - g(x + 5) ......(1)

Now

\int\limits_0^x {f(t)dt}

=

- \int\limits_0^x {g\left( {t + 5} \right)} dt

=

- \int\limits_5^{x + 5} {g\left( t \right)} dt

=

\int\limits_{x + 5}^5 {g(t)dt}

Step-by-Step Verified Solution