let f(x) = ∫_0 ^1 (dt/(2+ch(xt))) 1) find a explicit form of f(x) 2) calculate g(x)=∫_0 ^1 ((tsh(xt))/((2+ch(xt))^2 ))dt 3) find the value of ∫_0 ^1 (dt/(2+ch(3t))) and ∫_0 ^1 ((tsh(3t))/((2+ch(3t))^2 ))dt 4) calculate u_n =∫_0 ^1 (dt/(2+ch(nt))) with n natural integr and study the convergence of the serie Σ (u


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Question Number 47060 by maxmathsup by imad last updated on 04/Nov/18

$l e t f (x) = \int_{0}^{1} \frac{d t}{2 + c h (x t)}$ $1) f i n d a e x p l i c i t f o r m o f f (x)$ $2) c a l c u l a t e g (x) = \int_{0}^{1} \frac{t s h (x t)}{{(2 + c h (x t))}^{2}} d t$ $3) f i n d t h e v a l u e o f \int_{0}^{1} \frac{d t}{2 + c h (3 t)} a n d \int_{0}^{1} \frac{t s h (3 t)}{{(2 + c h (3 t))}^{2}} d t$ $4) c a l c u l a t e u_{n} = \int_{0}^{1} \frac{d t}{2 + c h (n t)} w i t h n n a t u r a l i n t e g r a n d s t u d y t h e c o n v e r g e n c e$ $o f t h e s e r i e Σ \frac{u_{n}}{n} .$
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