let f(x) =∫_0 ^∞ (dt/((x^2 +t^2 )^2 )) with x>0
1) find a explicit form of (x)
2)find also g(x) =∫_0 ^∞ (dt/((x^2 +t^2 )^3 ))
3)find the values of integrals ∫_0 ^∞ (dt/((t^2 +3)^2 )) and ∫_0 ^∞ (dt/((t^2 +3)^3 ))
4) calculate U_θ =∫_0 ^∞ (dt/((t^2 +cos^2 θ)^2 )) with 0<θ<(π/2)
5) find f^((n)) (x) and f^((n)) (0)
6) developp f at integr serie
Let consider an integer serie {a_n x^n } given by a_n = H_n =Σ_(k=1) ^n (1/k)
1) Find out the largest domain D of convergence of that integer serie
2) ∀ x∈D , explicit the sum S(x) of the {a_n x^n }
3) Calculate ∫_(−1) ^1 S(1−x)S(x) dx .
let f(x) =∫_0 ^2 (√(x+t^2 ))dt with x≥0
1) calculate f(x)
2)calculate g(x) =∫_0 ^2 (dt/(√(x+t^2 )))
3)find the value[of ∫_0 ^2 (√(4+t^2 ))dt and ∫_0 ^2 (dt/(√(3+t^2 )))
4) give g^′ (x) at form of integral.