How to find inverse of convolution integral?
The integral of a signal $x(t)$

$$f(t)=\int_{-\infty}^tx(\tau)d\tau$$

can be written as the convolution of $x(t)$ with the unit step function $u(t)$:

$$f(t)=(x\star u)(t)$$

Consequently, the integral of a convolution

$$g(t)=\int_{-\infty}^t(x\star y)(\tau)d\tau$$

can be written as

$$g(t)=((x\star y)\star u)(t)$$

And due to associativity of convolution we have

$$g(t)=((x\star u)\star y)(t)=((y\star u)\star x)(t)$$

So options $(c)$ and $(d)$ in your question are both correct.