Question

Im super confused as to how to approach this question?

Answer 1

1. -22
2. 20

Explanation:

We know that `int_a^b = -int_b^a`. Thus `int_(2)^6 f(x) dx = -int_6^2f(x)dx`. Since `int_2^6 f(x)dx = 16`, `int_6^2 f(x) dx =-16`

Also recall that

`int_a^bf(x) dx+ int_b^c f(x) dx = int_a^cf(x)dx`

Thus

`int_(-2)^6 f(x) dx + int_6^2 f(x)dx= int_(-2)^2 f(x)dx`

`-6 - 16 = int_(-2)^2f(x)dx`

`int_(-2)^2 f(x)dx =-22`

As for the second, we can expand and say

`int_(-2)^6[(g(x) + 2]dx = int_(-2)^6 g(x) dx + int_(-2)^6 2 dx`

`int_(-2)^6[g(x) + 2]dx = 4 + [2x]_(-2)^6`

`int_(-2)^6 [g(x) + 2]dx = 4 + 12- (2(-2))`

`int_(-2)^6 [g(x) + 2]dx = 20`

Hopefully this helps!