If I= displaystyle ∫ ((ln x)5/√x2 + x2 (ln â¡ x)3)dx= k√(ln x)3 + 1((ln â¡ x)3 - 2)+c (where c is the constant of integration), then 9k is equal to

Question

If I= displaystyle ∫ ((ln x)5/√x2 + x2 (ln â¡ x)3)dx= k√(ln x)3 + 1((ln â¡ x)3 - 2)+c (where c is the constant of integration), then 9k is equal to (A) 4 (B) 2 (C) 6 (D) 10

Tardigrade Admin · Accepted Answer

Let 1+(ln x)3=t2 ⇒ I= displaystyle ∫ ((t2 - 1) 2 t d t/3 t) ⇒ I=(2/3) displaystyle ∫ (t2 - 1)dt =(2 t3/9)-(2/3)t+c =(2/9)t(t2 - 3)+c =(2/9)√1 + (ln x)3((ln â¡ x)3 - 2)+c ∴ k=(2/9)⇒ 9k=2

Q. If $I = \int \frac{( l n x ) ^{5}}{x ^{2} + x ^{2} ( l n x ) ^{3}} d x =$ $k (l n x)^{3} + 1 ((l n x)^{3} - 2) + c$ (where $c$ is the constant of integration), then $9 k$ is equal to

$4$

$2$

$6$

$10$

Q. If I=∫x2+x2(ln⁡x)3​(lnx)5​dx= k(lnx)3+1​((ln⁡x)3−2)+c (where c is the constant of integration), then 9k is equal to

4

2

6

10

Let 1+(lnx)3=t2 ⇒I=∫3t(t2−1)2tdt​ ⇒I=32​∫(t2−1)dt =92t3​−32​t+c =92​t(t2−3)+c =92​1+(lnx)3​((ln⁡x)3−2)+c ∴k=92​⇒9k=2

Q. If $I = \int \frac{( l n x ) ^{5}}{x ^{2} + x ^{2} ( l n x ) ^{3}} d x =$ $k (l n x)^{3} + 1 ((l n x)^{3} - 2) + c$ (where $c$ is the constant of integration), then $9 k$ is equal to

$4$

$2$

$6$

$10$