Question

please helpsuppose that f is an even function, g is odd, both are integralable on [-5,5]…

Answer 1

Recall that the integral of an odd function h(x) over a symmetric interval is:

$\int ^a_{-a}h(x)dx=0._{}$

If h(x) is even then:

$\int ^a_{-a}h(x)dx=2\int ^a_0h(x)dx\text{.}$

Also, recall that:

$\int ^a_{-a}(h(x)+m(x))dx=\int ^a_{-a}h(x)dx+\int ^a_{-a}m(x)dx\text{.}$

Since f(x) is an even function, and g(x) is an odd function we get:

$\int ^5_{-5}(f(x)+g(x))dx=\int ^5_{-5}(f(x))dx+\int ^5_{-5}(g(x))dx=2\int ^5_0f(x)dx+0.$

Therefore:

$\int ^5_{-5}(f(x)+g(x))dx=2\times19=38.$

Answer:

$\int ^5_{-5}(f(x)+g(x))dx=38.$