If ∫ ( cos x - sin x +1- x / e x + sin x + x ) dx = ln (f( x ))+ g ( x )+ C where C is the constant of integration and f( x ) is positive, then f( x )+g( x ) has the value equal to

Question

If ∫ ( cos x - sin x +1- x / e x + sin x + x ) dx = ln (f( x ))+ g ( x )+ C where C is the constant of integration and f( x ) is positive, then f( x )+g( x ) has the value equal to (A) e x + sin x +2 x (B) e x + sin x (C) e x - sin x (D) e x + sin x + x

Tardigrade Admin · Accepted Answer

I=∫ (( e x + cos x +1)-( e x + sin x + x )/ e x + sin x + x ) dx = ln ( e x + sin x + x )- x + C ∴ f ( x )= e x + sin x + x text and g ( x )=- x f ( x )+ g ( x )= e x + sin x

Q. If $\int \frac{c o s x - s i n x + 1 - x}{e ^{x} + s i n x + x} d x = ln (f (x)) + g (x) + C$ where $C$ is the constant of integration and $f (x)$ is positive, then $f (x) + g (x)$ has the value equal to

$e^{x} + sin x + 2 x$

$e^{x} + sin x$

$e^{x} - sin x$

$e^{x} + sin x + x$

$I = \int \frac{( e ^{x} + c o s x + 1 ) - ( e ^{x} + s i n x + x )}{e ^{x} + s i n x + x} d x$
$= ln (e^{x} + sin x + x) - x + C$
$∴ f (x) = e^{x} + sin x + x and g (x) = - x$
$f (x) + g (x) = e^{x} + sin x$

Q. If ∫ex+sinx+xcosx−sinx+1−x​dx=ln(f(x))+g(x)+C where C is the constant of integration and f(x) is positive, then f(x)+g(x) has the value equal to

ex+sinx+2x

ex+sinx

ex−sinx

ex+sinx+x