Let $f\,\left( x \right)$ be a function satisfying $f'\left( x \right) = f\left( x \right)$ with $f\left( 0 \right) = 1$ and ${\rm{g}}\,\,\left( x \right)$ be a function that satisfies $f\left( x \right)\,\, + \,\,{\rm{g}}\left( x \right) = {x^2}$. Then, the value of the integral $\mathop \smallint \nolimits_0^1 \,\,f\left( x \right)\,\,{\rm{g}}\,\,\left( x \right)\,\,dx$, is

$e - \frac{{{e^2}}}{2} - \frac{5}{2}$

$e\,\, + \,\,\frac{{{e^2}}}{2}\,\, + \,\,\frac{5}{2}$

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

1 Hard

1 of 1

Application +4 | -1

Let

f\,\left( x \right)

be a function satisfying

f'\left( x \right) = f\left( x \right)

with

f\left( 0 \right) = 1

and

\mathrm{g}\,\,\left( x \right)

be a function that satisfies

f\left( x \right)\,\, + \,\,\mathrm{g}\left( x \right) = {x^2}

. Then, the value of the integral

\int \nolimits_0^1 \,\,f\left( x \right)\,\,\mathrm{g}\,\,\left( x \right)\,\,dx

, is

View answer and explanation

Correct option: C

e\,\, - \,\,\frac{{{e^2}}}{2}\,\, - \,\,\frac{3}{2}

Option C (

e\, - \,\frac{{{e^2}}}{2}\, - \,\frac{3}{2}

) is correct because it is the result of integrating the product of the two functions after finding f(x) and g(x).