Definite Integration Question 411
Question: The value of $ \int_1^{e^{2}}{\frac{dx}{x{{(1+\ln x)}^{2}}}} $ is
[J & K 2005]
Options:
A) $ 2/3 $
B) $ 1/3 $
C) $ 3/2 $
D) $ \ln 2 $
Show Answer
Answer:
Correct Answer: A
Solution:
$ I=\int_1^{e^{2}}{\frac{dx}{x{{(1+\ln x)}^{2}}}} $
Let $ (1+\ln x)=t $
therefore $ dt=\frac{1}{x}dx $
Now, when $ x=1\to e^{2} $ , then $ t=1\to 3 $
$ \therefore $ $ I=\int_1^{3}{\frac{dt}{t^{2}}=[ \frac{-1}{t} ]_1^{3}=-[ \frac{1}{3}-1 ]}=\frac{2}{3} $ .
BETA
