QJEEM21Ma4d11: JEE Main 2021 Mathematics - Class 12 JEEM21Ma4d11 Solution: Let f : R R be defined as f(x) = e x sinx. If F : [0, 1] R is a …

Question

Verified Solution • 2025–26 Session moderate

Let f : R

\to

R be defined as f(x) = e^{$-$ x}sinx. If F : [0, 1]

\to

R is a differentiable function with that F(x) =

\int_0^x {f(t)dt}

, then the value of

\int_0^1 {(F'(x) + f(x)){e^x}dx}

lies in the interval

Accepted Answer

F(x) =

\int_0^x {f(t)dt}

\Rightarrow

F'(x) = f(x) by Leibnitz theorem

I =

\int\limits_0^1 {(F'(x) + f(x)){e^x}dx = \int\limits_0^1 {2f(x){e^x}dx} }

I = \int\limits_0^1 {2\sin x\,dx}

I = 2(1 - \cos 1)

= 2\left\{ {1 - \left( {1 - {{{1^2}} \over {2!}} + {{{1^4}} \over {4!}} - {1 \over {6!}} + ...} \right)} \right\}

= 2\left\{ {1 - \left( {1 - {1 \over 2} + {1 \over {24}}} \right)} \right\} &lt; 2(1 - \cos 1) &lt; 2\left\{ {1 - \left( {1 - {1 \over 2} + {1 \over {24}} - {1 \over {720}}} \right)} \right\}

{{330} \over {360}} &lt; 2(1 - \cos 1) &lt; {{331} \over {360}}

{{330} \over {360}} &lt; 1 &lt; {{331} \over {360}}

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