Mathematics

# Evaluate the following integral$\int { \cfrac { 1-\cot { x } }{ 1+\cot { x } } } dx$

##### SOLUTION
$I=\displaystyle\int{\dfrac{1-\cot{x}}{1+\cot{x}}dx}$

$I=\displaystyle\int{\dfrac{1-\dfrac{\cos{x}}{\sin{x}}}{1+\dfrac{\cos{x}}{\sin{x}}}dx}$

$I=\displaystyle\int{\dfrac{\sin{x}-\cos{x}}{\sin{x}+\cos{x}}dx}$

Let $t=\sin{x}+\cos{x}\Rightarrow\,dt=\left(\cos{x}-\sin{x}\right)dx$

$\Rightarrow\,dt=-\left(\sin{x}-\cos{x}\right)dx$

$I=-\displaystyle\int{\dfrac{dt}{t}}$

$I=-\log{\left|t\right|}+c$

$\therefore\,I=-\log{\left|\sin{x}+\cos{x}\right|}+c$ where $t=\sin{x}+\cos{x}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle\lim_{n\rightarrow \infty}\dfrac{3}{n}\left\{1+\sqrt{\dfrac{n}{n+3}}+\sqrt{\dfrac{n}{n+6}}+\sqrt{\dfrac{n}{n+9}}+...….+\sqrt{\dfrac{n}{n+3(n-1)}}\right\}=?$
• A. Does not exist
• B. $1$
• C. $3$
• D. $2$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Find $\displaystyle\int _{ 0 }^{ 5 }{ \left( x+1 \right) dx }$ as limit of a sum.

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
$\displaystyle \int_{-\pi}^{\pi}(\cos px-\sin qx)^{2} dx$ ,where $p,q$ are integers is equal to
• A. $-\pi$
• B.
• C. $\pi$
• D. $2\pi$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
lf $f(x)=\displaystyle \frac{x+2}{2x+3}$ , then $\displaystyle \int(\frac{f(x)}{x^{2}})^{1/2}dx$ is

equal to
$\displaystyle \frac{1}{\sqrt{2}}g(\frac{1+\sqrt{2f(x)}}{1-\sqrt{2f(x)}})-\sqrt{\frac{2}{3}}h(\frac{\sqrt{3f(x)}+\sqrt{2}}{\sqrt{3f(x)}-\sqrt{2}})+c$
where
• A. $g(x)=\tan^{-1}x,\ h(x)=\log|x|$
• B. $g(x)=\log|x|,h(x)=\tan^{-1}x$
• C. $g(x)=h(x)=\tan^{-1}x$
• D. $g(x)=\log|x|, h(x)=\log|x|$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
If $n\rightarrow \infty$ then the limit of series in $n$ can be evaluated by following the rule : $\displaystyle \lim_{n\rightarrow \infty}\sum_{r=an+b}^{cn+d}\frac{1}{n}f\left ( \frac{r}{n} \right )=\int_{a}^{c}f(x)dx,$
where in $LHS$, $\dfrac{r}{n}$ is replaced by $x$,
$\dfrac{1}{n}$ by $dx$
and the lower and upper limits are $\lim_{n\rightarrow \infty }\dfrac{an+b}{n}\, and \, \lim_{n\rightarrow \infty }\dfrac{cn+d}{n}$ respectively.