∫ ( ln | x |/ x √1+ ln | x |) dx equals :

Question

∫ ( ln | x |/ x √1+ ln | x |) dx equals : (A) (2/3) √1+ ln | x |( ln | x |-2)+ c (B) (2/3) √1+ ln | x |( ln | x |+2)+ c (C) (1/3) √1+ ln | x |( ln | x |-2)+ c (D) 2 √1+ ln | x |(3 ln | x |-2)+ c

Tardigrade Admin · Accepted Answer

operatornameStart ln | x |= t ⇒ (1/ x ) dx = dt ⇒ ∫ ( tdt /√1+ t )

Q. $\int \frac{l n ∣ x ∣}{x 1 + l n ∣ x ∣} d x$ equals :

$\frac{2}{3} 1 + ln ∣ x ∣ (ln ∣ x ∣ - 2) + c$

$\frac{2}{3} 1 + ln ∣ x ∣ (ln ∣ x ∣ + 2) + c$

$\frac{1}{3} 1 + ln ∣ x ∣ (ln ∣ x ∣ - 2) + c$

$2 1 + ln ∣ x ∣ (3 ln ∣ x ∣ - 2) + c$

$Start ln ∣ x ∣ = t \Rightarrow \frac{1}{x} d x = d t \Rightarrow \int \frac{t d t}{1 + t}$

Q. ∫x1+ln∣x∣​ln∣x∣​dx equals :

32​1+ln∣x∣​(ln∣x∣−2)+c

32​1+ln∣x∣​(ln∣x∣+2)+c

31​1+ln∣x∣​(ln∣x∣−2)+c

21+ln∣x∣​(3ln∣x∣−2)+c