Mathematics

# Evaluate the following integral:$\displaystyle \int { co\sec { x } \log { \left( co\sec { x } -\cot { x } \right) } } dx\quad$

##### SOLUTION
We've,
$\dfrac{d}{dx}\{\log (cosec x-\cot x\}=\dfrac{-cosec x.\cot x+cosec^2 x}{cosec x-\cot x}=cosec x.$.....(1).

Now,

$\int { cosec { x } .\log { \left( cosec { x } -\cot { x } \right) } } dx$

$=\int { \log { \left( cosec { x } -\cot { x } \right) } } d\{\log(cosec x-\cot x)\}$ [ Using (1)]

$=\dfrac{\{\log(cosec x-\cot x)\}^2}{2}+c$. [ Where $c$ is integrating constant]

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium

$\displaystyle \int_{0}^{1}\sin^{-1}(\frac{2x}{1+x^{2}})dx=$
• A. $\displaystyle \frac{\pi}{4}$
• B. $\displaystyle \frac{\pi}{4}+\log 2$
• C. $\displaystyle \frac{\pi}{2}+\frac{1}{2}$ log2
• D. $\displaystyle \frac{\pi}{2}-\log 2$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve: $\int \sin^32x.dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\displaystyle\int_{0}^{2} 3x+2\ dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\displaystyle\int{\frac{dx}{(x^2+1)(x^2+4)}}=k\tan^{-1}{x}+l\tan^{-1}{\frac{x}{2}}+C$, then
• A. $\displaystyle k=\frac{1}{3}$
• B. $\displaystyle l=\frac{2}{3}$
• C. $\displaystyle l=-\frac{1}{6}$
• D. $\displaystyle k=-\frac{1}{3}$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.