Finding mean value of a continuous function

Question

Assignment:

Answers and work for Integration questions:

Q1.Integrate the following functions with respect to Θ.

(i) sin(5Θ - 4)
(ii) cos(3 - 2Θ)

Q2. Integrate the following functions with respect to x.

(i) 4e-3x
(ii) (3 + 4x)-1
(iii) -6 / 2 – 5x
(iv) 5 – 3e4x /e^x

Q3. Evaluate the following definite integrals.
       _π/3
(i)  ∫     cos(2Θ)dΘ
      ⁰
      _{0. 5}
(ii) ∫     sin(1.6 + 3Θ)dΘ
      ^0.1
                        ₁
Q4. Evaluate  ∫ (3 – e_4x) dx
                       ⁰
                       ₄
Q5. Evaluate  ∫   (1 / 1 + 4x) dx
                       ³ 
                       ₀
Q6. Evaluate  ∫  (5 /3 – 2x)  dx
                      ^-1      
                      _-2
7. Evaluate   ∫    (e_3x – 4 / e_x)  dx   
                      ^-3          

Q8. The mean value of a continuous function over a given range is defined as the integral of the function divided by the range.
                                                                        _b
Mean value of f(x) from a to b =    1 /b – a   ∫     f(x)dx
                                                                        ^a
Determine the mean value of the function 1 – e_-t/T from t = 0 to t = T.

Provide complete and step by step solution for the question and show calculations and use formulas.