Experts please help with probability density function

Really appreciate if any experts can help with this question?

If,

fx(x) { cx^3 if 1<= x <=2
0 otherwise

1) What does c need to take for fx to be valid probability desity function?

2) Find the distribution function, fx(x) = P(X <= x), of X

3) What is the probability that X > 1.5?

1) We need ∫(x = 1 to 2) cx^3 dx = 1
==> (1/4)cx^4 {for x = 1 to 2} = 1
==> c = 4/15.

2) For x ≤ 1, Fx(x) = 0.
For x > 2, Fx(x) = 1.

Otherwise,
Fx(x) = ∫(t = 1 to x) (4/15)t^3 dt
.........= (1/15)t^4 {for t = 1 to x}
.........= (1/15)(x^4 - 1).
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3) P(X > 1.5)
= 1 - P(X < 1.5)
= 1 - (1/15)(1.5^4 - 1), using part b
= 35/48.

I hope this helps!