Getting Ready for the Final - Part 1 1. If ∈t _1^3f(w)dw=7 , find the value of ∈t _1^2f(5-2x)dx.

2 Solve for dx to get $dx = \frac{du}{-2}$

3 Change the limits of integration when substituting u for 5 - 2x . When x = 1 , u = 5 - 2(1) = 3 , and when x = 2 , u = 5 - 2(2) = 1

4 Rewrite the integral in terms of u :

5 Since $\frac{1}{-2}$ is a constant, we can take it out of the integral:

6 Since $$\int_{3}^{1} f(u) \, du = \int_{1}^{3} f(w) \, dw$$ (by the property of definite integrals), we have:

7 Calculate the final value: