Integral Question

Question

Let ∫[−4.5 to −9] f(x) dx= 5, ∫ [−7.5 to −9] f(x) dx= 2, ∫[−4.5 to −6] f(x) dx= 4.

Find ∫[−6 to −7.5] f(x) dx=
and
∫ [−7.5 to −6] 5 * f(x) −2 dx=

I found out that ∫[−6 to −7.5] f(x) dx= -1
however the second part is confusing me.

Answer 1

I think the integrand is 5f(x) - 2 for ∫ [−7.5 to −6] 5 * f(x) − 2 dx.  We can then use integration rules to break up the integral into 2 parts:

 

 

 ∫[−7.5 to −6] (5f(x) − 2) dx

 

 

=  5∫ [−7.5 to −6]f(x) dx − 2∫ [−7.5 to −6] dx 

 

 

= 5(1) - 2(-7.5 - (-6)) = 5 - 2(-1.5) = 8

 

 

We have used ∫[−6 to −7.5] f(x) dx= -∫[−7.5 to −6] f(x) dx = -1.