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$\lim_{n\rightarrow \infty }\left ( \frac{n}{n^{2}+1^{2}}+\frac{n}{n^{2}+2^{2}}+\frac{n}{n^{2}+3^{2}}+\cdots +\frac{1}{5n} \right )$ is equal to :
Option 1)
$\tan ^{-1}\left ( 3 \right )$

Option 2)

$\frac{\pi }{2}$
Option 3)

$\tan ^{-1}\left ( 2 \right )$
Option 4)

$\frac{\pi }{4}$

Answers (1)

best_answer

Integration of Rational and irrational function -

Integration in the form of :

(i) $\int \frac{dx}{x^{2}+a^{2}}$ (ii) $\int \frac{dx}{x^{2}-a^{2}}$ (iii) $\int \frac{dx}{a^{2}-x^{2}}$

(iv) $\int \frac{dx}{\sqrt{x^{2}+a^{2}}}$ (v) $\frac{dx}{\sqrt{x^{2}-a^{2}}}$ (vi) $\frac{dx}{\sqrt{a^{2}-x^{2}}}$

- wherein

(i) $\frac{1}{a}tan^{-1}\, \frac{x}{a}+C$

(ii) $\frac{1}{2a}log \left | \frac{x-a}{x+a} \right |+C$

(iii) $\frac{1}{2a}log \left | \frac{x+a}{x-a} \right |+C$

(iv) $log\left | x+ \sqrt{x^{2}+a^{2}} \right |+C$

(v) $log\left | x+ \sqrt{x^{2}-a^{2}} \right |+C$

(vi) $sin^{-1} \left(\frac{x}{a} \right )+C$

6

$\lim_{n\rightarrow \infty }\sum_{r=1}^{2n}\frac{n}{n^{2}+r^{2}}\\\\\int_{0}^{2}\frac{dx}{1+x^{2}}=tan^{-1}2$

Option 1)

$\tan ^{-1}\left ( 3 \right )$

Option 2)

$\frac{\pi }{2}$

Option 3)

$\tan ^{-1}\left ( 2 \right )$
Option 4)

$\frac{\pi }{4}$

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