let f(x)=∫_0 ^1 ((ln(1+xt^2 ))/(1+t^2 ))dt
1) find a xplicit form of f(x)
2) developp f at integr serie
3)find the value of ∫_0 ^1 ((ln(1+t^2 ))/(1+t^2 ))dt
4)find the value of ∫_0 ^1 ((ln(1+2t^2 ))/(1+t^2 ))dt
let f(x)=∫_0 ^(2π) ((sin(2t))/(1+x cos(t)))dt
1) find a explicit form of f(x)
2) find also g(x)=∫_0 ^(2π) ((sin(2t)cost)/((1+xcost)^2 ))dt
3)find the value of ∫_0 ^(2π) ((sin(2t))/(1+3 cos(t)))dt and
∫_0 ^(2π) ((cost sin(2t))/((1+3cost)^2 ))dt .
let f(x) =∫_(1/2) ^1 (dt/(2+ch(xt)))
1) find a explicit form of f(x)
2) calculate g(x)=∫_(1/2) ^1 ((tsh(xt))/((2+ch(xt))^2 ))dt
3) find the value of ∫_(1/2) ^1 (dt/(2+ch(3t))) and ∫_(1/2) ^1 ((tsh(2t))/((2+ch(2t))^2 ))dt
4) let u_n =∫_(1/2) ^1 (dt/(2+ch(nt))) study the convergence of Σu_n
and Σ(u_n /n) .
let W(x) =∫_(−∞) ^(+∞) ((arctan(xt^2 ))/(2+t^2 ))dt
1) find a explicit form of f(x)
2) find the value of ∫_(−∞) ^(+∞) (t^2 /((2+t^2 )(1+x^2 t^4 )))dt .