Question

How do you evaluate the definite integral `int (x^4+2x^2-5)dx` from [-2,2]?

Answer 1

`int_(-2)^2 (x^4+2x^2-5)dx = 52/15`

Explanation:

We can note that:

`f(x) = x^4+2x^2-5`

is an even function as `f(-x) = f(x)`, so:

`int_(-2)^2 (x^4+2x^2-5)dx = 2int_0^2 (x^4+2x^2-5)dx`

`int_(-2)^2 (x^4+2x^2-5)dx = 2[x^5/5+(2x^3)/3-5x]_0^2 = 2(32/5+16/3-10)=2((96+80-150)/15)=52/15`