Integration and Fundamental Theorem Calculus

The solution file is attached.

Question 1
Find
∫ x3+4
________________________________________x2 dx

 (x^3 + 4)dx /x^2 =  (x^3 /x^2) dx + 4/(x^2) dx =  x dx + 4  (1/x^2)dx
= x^2 /2 - (4/x) + C

Question 2

Apply the Fundamental Theorem of Calculus to find the derivative of:
h(x)= x∫2^/¯u-1dx

I am not sure how to interpret the integral here. Let me do it in two different ways ...
(a)  [Öu - 1]dx =  Öu dx -  dx = Öu * (x) - x + C = x(Öu - 1) + C
On applying the limits, we get the integral as x(Öu - 1) - 2(Öu - 1) = (Öu - 1)(x - 2)
(b)  Ö(u - 1) dx = Ö(u - 1)  dx = [Ö(u - 1)] x + C
On applying the limits, we get the integral as [Ö(u - 1)](x - 2)

Question 3

Evaluate:
∫2cos2 xdx

 2 cos^2 x dx =  (1 + cos 2x) dx =  dx +  cos 2x dx = x + (1/2) sin 2x + C.

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