Question

Evaluate the integral :
(i) ` int_(1)^(5) x^(2)dx " "` (ii) ` int_(0)^(1) t^(2)dt" "` (iii) `int_(3)^(5) t dx " "` (iv) ` int_(0)^(1) t^(3//2)dx`

Answer 1

(i) `underset(1)overset(5)intx^(2)dx=[(x^(3))/(3)]_(1)^(5)=(1)/(3)[x^(3)]_(1)^(5)=(1)/(3)((5^(3))-(1)^(3))=(1)/(3)(125-1)=(124)/(3)`
(ii) `underset(0)overset(1)intt^(2)dt=[(t^(3))/(3)]_(0)^(1)=(1)/(3) " " (iii) underset(3)overset(5)int t dt =[(t^(2))/(2)]_(3)^(5)=(5^(2)-3^(2))/(2) =8`
(iv)`underset(0)overset(1)intt^(3//2) dx=[t^(5//2)/(5//2)]_(0)^(1)=(2)/(5)`