Even Odd Functions

5.6

• Even and odd functions

  • if a function is even

    • ${\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx$

  • if a function is odd

    • ${\int }_{-a}^{a}f(x)dx=0$

  • Ex.

    • a. ${\int }_3^3(2x^4+3x^2+4)dx$

      • ⇒ $2{\int }_0^3(2x^4+3x^2+4)dx$
        • this makes it much easier as we can use just zero so we dont need to calculate the lower bound

    • b. ${\int }_e^e\frac{e^{u^2}\sin u}{u^2+10}du$

      • ⇒ since its an even function being times by an odd the top of the fraction is odd then since its divided by even its still odd
        • ⇒ since integrand is odd we know that its 0

  • How to find if its even or odd

    • $f_1,f_2,f_3$ these are odd but $f_4$ is even, we can think of it of it all being that all of the odd f are negatives and when we multiply with the even function it still is odd! same thing with division