Question

Suppose that f is an even function, g is odd, both are integrable on [-5,5]..

Answer 1

Since f(x) is even, we have:

`\int ^5_(-5)f(x)dx=2\int ^5_0f(x)dx`

And since g(x) is odd, we have:

`\int ^5_(-5)g(x)dx=0`

Now, using those results, we obtain:

`\int ^5_(-5)\lbrack f(x)+g(x)\rbrack dx=\int ^5_(-5)f(x)dx+\int ^5_(-5)g(x)dx=2\int ^5_0f(x)dx+0=2\int ^5_0f(x)dx`

And since

`\int ^5_0f(x)dx=19`

we obtain:

`\int ^5_(-5)\lbrack f(x)+g(x)\rbrack dx=2\cdot19=38`