Question

If $f$ is even or odd function then

Answer 1

Using Properties
${\int }_{-a}^{2}g\left(x\right)dx=2{\int }_0^{a}g\left(x\right)dx$ ...(1)
And ${\int }_a^{b}g\left(x\right)dx={\int }_a^{b}g(a+b-x)dx$ ...(2)
A) Using (1) ${\int }_{-a}^a{\left(f\left(x\right)\right)}^{2}dx=2{\int }_0^a{\left(f\left(x\right)\right)}^{2}dx$
B) Let $I={\int }_{-a}^{a}x(f{\left(x\right)}^{2}dx$ ...(3)
From (2) $I=-{\int }_{-a}^{a}x(f{\left(x\right)}^{2}dx$ (4)
From (3) and (4) we get $2I=0\Rightarrow I=0$
C) Let $J={\int }_{-a}^a(x^2+x^3){\left(f\left(x\right)\right)}^{2}dx$ ...(5)
From (2) $J={\int }_{-a}^a(x^2-x^3){\left(f\left(x\right)\right)}^{2}dx$ ...(6)
From (5) and (6)
$2J={\int }_{-a}^a{x}^2{\left(f\left(x\right)\right)}^{2}dx$
Now using (1) $2J=2{\int }_0^a{x}^2{\left(f\left(x\right)\right)}^{2}dx\Rightarrow J={\int }_0^a{x}^2{\left(f\left(x\right)\right)}^{2}dx$
D) Let $K={\int }_{-a}^a{\left(f\left(x\right)\right)}^{4}dx=2{\int }_0^a{\left(f\left(x\right)\right)}^{4}dx$ from (1)
And from (2) $K=2{\int }_0^a{\left(f(a-x)\right)}^{4}dx$
Therefore
$2K=2{\int }_0^a{\left(f\left(x\right)\right)}^{4}dx+2{\int }_0^a{\left(f(a-x)\right)}^{4}dx\Rightarrow K={\int }_0^a{\left(f\left(x\right)\right)}^{4}dx+{\int }_0^a{\left(f(a-x)\right)}^{4}dx$