evaluation of chi-squared density

Question

I am trying to work out the cdf of the chi-squared distribution. My query is whether it is possible to evaluate this integral:

$$\int_0^y t^{n/2-1} e^{-t/2} \mathrm{d}t$$

Answer 1

Make $t=2x$ $$\int_0^y t^{\frac n2-1} e^{-\frac t2} \, dt=2^{\frac n2}\int_0^{\frac y 2} x^{\frac{n}{2}-1}e^{-x}\,dx$$ and use the gamma function.

$$\int_0^y t^{\frac n2-1} e^{-\frac t2} \, dt=2^{\frac n2} \left(\Gamma \left(\frac{n}{2},0\right)-\Gamma \left(\frac{n}{2},\frac{y}{2}\right)\right)$$