Convolution of two uniform

Let $$X$$ be a uniform random variable on $$[0,1]$$, let $$Y$$ be uniform in $$[3,5]$$ independent of $$X$$. Find the probability density function of $$X + Y$$.

My solution is: $$(f_X * f_Y)(x) = \begin{cases} \displaystyle \int_{0}^{x-3} f_X(y) f_Y(x-y)\,dy & \text{if }3

You are not finished !!! You have used your knowledge that $$f_X$$ is zero outside of $$[0,1]$$. But you also know its value inside $$[0,1]$$... Same for $$f_Y$$. Substitute the values and compute the integrals.