Question

用函数的奇偶性求下列积分

第四大题的三个小题

Answer 1

如图所示

Answer 2

(1)
f(x) = | arctanx|
f(-x) = f(x)
∫(-√3->√3) | arctanx| dx
=2∫(0->√3) | arctanx| dx
=2∫(0->√3) arctanx dx
=2[xarctanx]|(0->√3) - 2∫(0->√3) x/(1+x^2) dx
=2√3(π/3) - [ln|1+x^2|]|(0->√3)
=2√3(π/3)-2ln2
(2)
f(x)= x^3.cosx
f(-x)= -f(x)
∫(-π->π) ( x^3.cosx -1) dx
=∫(-π->π) x^3.cosx -∫(-π->π) dx
=0 -∫(-π->π) dx
=-2π
(3)
f(x)= x/(1+x^2)
f(-x)=-f(x)
g(x) = 1/(1+x^2)
g(-x) =g(x)
∫(-1->1) (x+1)/(1+x^2) dx
=∫(-1->1) x/(1+x^2) dx +∫(-1->1) dx/(1+x^2)
=0 +2∫(0->1) dx/(1+x^2)
=2[arctanx]|(0->1)
=π/2