Mathematics

# $\displaystyle \lim_{n\rightarrow \infty }\left [ \frac{1}{n^{2}}\sec ^{2}\frac{1}{n^{2}}+\frac{2}{n^{2}}\sec ^{2}\frac{4}{n^{2}}+...+\frac{1}{n}\sec ^{2}1 \right ]$ equals

$\displaystyle \frac{1}{2}\tan 1$

##### SOLUTION
$\displaystyle \lim_{n\rightarrow \infty }\frac{1}{n^{2}}\sec ^{2}\frac{1}{n^{2}}+\frac{2}{n^{2}}\sec ^{2}\left ( \frac{4}{n^{2}} \right )+...+\frac{1}{n}\sec ^{2}1$
$\displaystyle =\lim_{n\rightarrow \infty }\frac{1}{n^{2}}\sec ^{2}\frac{1}{n^{2}}+\frac{2}{n^{2}}\sec ^{2}\left ( \frac{4}{n^{2}} \right )+...+\frac{n}{n^{2}}\sec ^{2}\left ( \frac{n^{2}}{n^{2}} \right )$
$\displaystyle =\lim_{n\rightarrow \infty }\sum_{r=1}^{r=n}\left ( \frac{r}{n^{2}} \right )\sec ^{2}\left ( \frac{r}{n} \right )^{2}=\lim_{n\rightarrow \infty }\sum_{r=1}^{r=n}\frac{1}{n}\left ( \frac{r}{n} \right )\sec ^{2}\left ( \frac{r}{n} \right )^{2}$
$\displaystyle =\int_{0}^{1}x\sec ^{2}\left ( x^{2} \right )dx=\frac{1}{2}\tan 1$
Hence, option 'C' is correct.

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Single Correct Medium

$\displaystyle \int_{0}^{\infty}\frac{x^{2}dx}{(1+x^{2})^{7/2}}=$
• A. 1/15
• B. -1/15
• C. $\displaystyle \frac{4}{15}$
• D. 2/15

1 Verified Answer | Published on 17th 09, 2020

Q2 One Word Medium
$\displaystyle\int \frac{1}{4\sin ^{2}x+9\cos ^{2}x}dx=\frac{1}{k}\tan ^{-1}\left ( \frac{2}{3}\tan x \right ).$ Find the value of $k$.

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Evaluate:
$\displaystyle\int \dfrac {\sin x\ dx}{\sin^{3}x+\cos^{3}x}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral as limit of sum:
$\displaystyle \int_{0}^{5}(x+1)dx$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Hard
In calculating a number of integrals we had to use the method of integration by parts several times in succession.
The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts
$\displaystyle \int u\left ( x \right )v\left ( x \right )dx=u\left ( x \right )v_{1}-u'\left ( x \right )v_{2}\left ( x \right )+u''\left ( x \right )v_{3}\left ( x \right )+...+\left ( -1 \right )^{n-1}u^{n-1}\left ( x \right )V_{n}\left ( x \right ) \\ -\left ( -1 \right )^{n-1}\int u^{n}\left ( x \right )V_{n}\left ( x \right )dx$
where  $\displaystyle v_{1}\left ( x \right )=\int v\left ( x \right )dx,v_{2}\left ( x \right )=\int v_{1}\left ( x \right )dx ..., v_{n}\left ( x \right )= \int v_{n-1}\left ( x \right )dx$
Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration by parts is especially useful when  calculating $\displaystyle \int P_{n}\left ( x \right )Q\left ( x \right )dx,$ where $\displaystyle P_{n}\left ( x \right )$ is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n+1 times.