From original burst, fraction of stellar mass still surviving on Main sequence Suppose that all stars in this galaxy were born in a single major-merger burst event about 10
Gyr ago. From this original burst, I want to compute the fraction of stellar mass still surviving as stars in the
main sequence ? For this, I have got to use a Salpeter IMF, and a star formation range between 0.1 and 120 solar
masses.

What I have done is starting from Salpeter IMF : $$\Phi(m)\text{d}m=\Phi_{0}\,m^{-2.35}$$

with $$\Phi_{0}$$ a constant normalization.

From this, I integrate from $$m_{1}=0.1\,\text{M}_{\odot}$$ to $$m_{2}=120\,\text{M}_{\odot}$$

$$N(0.1&lt;m&lt;120) = \int_{0.1}^{120}\,\Phi(m)\,\text{d}m = \Phi_{0}\,\bigg[\dfrac{0.1^{-1.35}-120^{-1.35}}{1.35}\bigg]$$

This result depends on the valeur of $$\Phi_{0}$$ and I don't know how to deal with it in order to get $$N(0.1&lt;m&lt;120)$$ ?

Moreover, it seems that I have to take into account of the age of the major-merger burst event (10 Gyr).

From these 2 principles, how could I calculate the fraction of stars surviving in the main sequence ?

Any help is wlecome, Regards
When you are calculating fractions, rather than absolute numbers, the value of $\Phi_0$ does not matter, since it will be a multiplying factor in both the numerator and denominator.

You have (almost, see below)  successfully got an expression for the denominator of your fraction.

The numerator is found by calculating an equivalent integral from your lower limit to an upper limit that is instead defined by the most long-lived main sequence stars that are still "alive" - i.e. those with a lifetime equal to the age of our Galaxy.

Finally, you were asked to find the fraction of stellar mass surviving, not the fraction of stars. The stellar mass existing between two mass intervals is
$$M_* = \int_{m_1}^{m_2} m\Phi(m)\ dm$$