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Question Number 45043 by maxmathsup by imad last updated on 07/Oct/18
let f(x)=((1+(√(1+x^2 )))/x) 1) calculate ∫_1 ^3 f(x)dx 2) determine f^(−1) (x) 3) find ∫ f^(−1) (x)dx .
letf(x)=1+1+x2x1)calculate∫13f(x)dx2)determinef−1(x)3)find∫f−1(x)dx.
Answered by MJS last updated on 08/Oct/18
∫((1+(√(x^2 +1)))/x)dx= [t=(√(x^2 +1)) → dx=((√(x^2 +1))/x)dt] =∫(t/(t−1))dt= [u=t−1 → dt=du] =∫((u+1)/u)du=u+ln u=t−1+ln(t−1)= =(√(x^2 +1))−1+ln∣(√(x^2 +1))−1∣= =ln∣1−(√(x^2 +1))∣+(√(x^2 +1))+C (1) ∫_1 ^3 f(x)dx=ln((√(10))+2(√5)−(√2)−1) +((√5)−1)(√2)≈3.40060
∫1+x2+1xdx=[t=x2+1→dx=x2+1xdt]=∫tt−1dt=[u=t−1→dt=du]=∫u+1udu=u+lnu=t−1+ln(t−1)==x2+1−1+ln∣x2+1−1∣==ln∣1−x2+1∣+x2+1+C(1)∫31f(x)dx=ln(10+25−2−1)+(5−1)2≈3.40060
Commented by maxmathsup by imad last updated on 08/Oct/18
thank you sir MJS.
thankyousirMJS.
Answered by MJS last updated on 08/Oct/18
(2) f(x) is defined for x≠0 range is R\[−1; 1] solving x=((1+(√(y^2 +1)))/y) gives y=((2x)/(x^2 −1)) but this is defined for x∈R\{−1; 1} especially for x∈]−1; 1[ ⇒ f^(−1) (x)=((2x)/(x^2 −1)) with x∈R\[−1; 1]
(2)f(x)isdefinedforx≠0rangeisR∖[−1;1]solvingx=1+y2+1ygivesy=2xx2−1butthisisdefinedforx∈R∖{−1;1}especiallyforx∈]−1;1[⇒f−1(x)=2xx2−1withx∈R∖[−1;1]
Answered by MJS last updated on 08/Oct/18
(3) ∫((2x)/(x^2 −1))dx= [t=x^2 −1 → dx=(dt/(2x))] =∫(dt/t)=ln t =ln ∣x^2 −1∣+C
(3)∫2xx2−1dx=[t=x2−1→dx=dt2x]=∫dtt=lnt=ln∣x2−1∣+C

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