Mathematics

# Evaluate the following definite integral:$\displaystyle \int _{\pi /4}^{\pi /2} \cot x \ dx$

##### SOLUTION
$\displaystyle \int^{\dfrac{\pi}{2}}_{\dfrac{\pi}{4}}cotxdx$

$=\displaystyle \int^{\dfrac{\pi}{2}}_{\dfrac{\pi}{4}}\dfrac{cosx}{sinx}dx$

$=ln|sinx| ^{\dfrac{\pi}{2}}_{\dfrac{\pi}{4}}$     $\left[\displaystyle \because \int{\dfrac{f'(x)}{f(x)}}dx=\log|f(x)|\right]$

$=ln(sin\dfrac{\pi}{2})-ln\left(sin\dfrac{\pi}{4}\right)$

$=ln(1)-ln\left(\dfrac{1}{\sqrt{2}}\right)$

$=\dfrac{1}{2}ln2$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Assertion & Reason Medium
##### ASSERTION

$\displaystyle \int_{a}^{x}f(t)dt$ is an even function if $f(x)$ is an odd function.

##### REASON

$\displaystyle \int_{a}^{x}f(t)dt$ is an odd function if $f(x)$ is an even function.

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1.
• B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1.
• C. STATEMENT-1 is False, STATEMENT-2 is True.
• D. STATEMENT-1 is True, STATEMENT-2 is False.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Evaluate $\displaystyle\int^{\pi/3}_0\tan x dx$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
$\int_{0}^{a}\dfrac{\sqrt{x}}{\sqrt{x}+\sqrt{a-x}}dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate: $\displaystyle \int \frac {(x^2+1)}{(x^4+1)}dx$

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.