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*19=38.`

Answer:

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