Intergrate the following functions w.r.t.x and find withinj the limits x = 2 to 4.
(i) `x^3` (ii) `sqrtx` (iii) `x^(-1)` (iv) `x^(-1//2)`

Answer

(i) 60 (ii) 3.448 (iii) 0.6931 (iv) `4 - 2 sqrt2`

Explanation

(i) `int_2^4 x^3 dx = ((x^4)/(4))_2^4 = (4^4)/(4) -(2^4)/(4) =60`
(ii) `int_2^4 sqrtx dx =int_2^4 x^(1//2) dx = ((x^(3//2))/(3//2))_(2)^(4)=2/3[4^(3//2) -2^(3//2)] = (2)/(3)[8-2sqrt2] = 3.448`
(iii) `int_2^4x^(-1) dx =int_2^4 (dx)/(x) =(log_e x)_2^4 = log_e 4 -log_e 2= log_e(4)/(2) = log_e2= 2.3026 log_(10)2`
`= 2.3026xx0.3010 = 0.6931`
(iv) `int_2^4 x ^(-1//2)dx = ((x^(-1)/(2)+1)/((-1)/(2) +1))_(2)^(4)=2(x^(1//2))_2^4 = 2[4^(1//2) - 2^(1//2)] = 2 (2-sqrt2) = 4-2sqrt2`