1. Differentiate the given integral identity

We are given

Since this holds for all $x>0$, the integrand must equal the derivative of the right-hand side:

2. Compute the derivative of $\dfrac{xg(x)}{e^x+1}$

Using the quotient rule,

Now,

So,

This must equal

Write the first term on RHS over the common denominator:

Hence RHS becomes

3. Equate numerators

Therefore,

Notice that the $g(x)$-terms on both sides cancel:

So we get

Since $x>0$ and $e^x+1>0$, divide by $x(e^x+1)$:

4. Find $g(x)$

Integrating,

where $C$ is a constant.

5. Check each option

Option A: $g$ is decreasing in $\left(0,\frac\pi4\right)$

We have

For $0<x<\frac\pi4$, we know

So $g$ is increasing, not decreasing.

Hence, A is false.

Option B: $g'$ is increasing in $\left(0,\frac\pi4\right)$

Differentiate again:

For $0<x<\frac\pi4$, both $\sin x,\cos x>0$, so

Thus $g'$ is decreasing, not increasing.

Hence, B is false.

Option C: $g+g'$ is increasing in $\left(0,\frac\pi2\right)$

Compute:

Differentiate:

For $0<x<\frac\pi2$,

So $g+g'$ is decreasing, not increasing.

Hence, C is false.

Option D: $g-g'$ is increasing in $\left(0,\frac\pi2\right)$

Compute:

Differentiate:

For $0<x<\frac\pi2$,

So $g-g'$ is increasing on $\left(0,\frac\pi2\right)$.

Hence, D is true.

6. Final answer

The only correct option is

7. Comparison with stored answer

Stored correct answer: $D$.

This matches our derived answer.