Mathematics

# Evaluate: $\displaystyle\int \sec ^{4}x. \mathrm{cosec} ^{2}xdx$

$\displaystyle \frac{1}{3}t^{3}+2t-\frac{1}{t}.$

##### SOLUTION
$\displaystyle I=\int \frac{1}{\cos ^{4}x\sin ^{2}x}dx=\int \frac{\left ( \sin ^{2}x+\cos ^{2}x \right )^{2}}{\cos ^{4}x\sin ^{2}x}dx$ $\displaystyle =\int \left [ \frac{\sin ^{2}x}{\cos ^{4}x}+\frac{1}{\sin ^{2}x} +\frac{2}{\cos ^{2}x}\right ]dx$ $\displaystyle =\int \left ( \sec ^{2}x\tan ^{2}x+co\sec ^{2}x+2\sec ^{2}x \right )dx=\frac{1}{3}\tan ^{3}x-\cot x+2\tan x$ Alt. Divide above and below by $\displaystyle \cos ^{4+2}x=\cos ^{6}x$ $\displaystyle \therefore =\int \frac{\sec ^{4}x,\sec ^{2}x}{\tan ^{2}x}dx=\int \frac{\left ( 1+\tan ^{2}x \right )^{2}}{\tan ^{2}x}\sec ^{2}xdx$ $\displaystyle =\int \frac{\left ( 1+t^{2} \right )^{2}}{t^{2}}dt=\int \left ( t^{2}+2+\frac{1}{t^{2}} \right )dt$ $\displaystyle =\frac{1}{3}t^{3}+2t-\frac{1}{t}etc.$

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

#### Realted Questions

Q1 Subjective Medium
$\displaystyle\int { \sqrt { 4-{ x }^{ 2 } } } dx$ is equals to

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve: $\displaystyle \int \dfrac{1}{\sqrt{1 - e^{2x}}} dx$

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$\int\limits_0^1 {\frac{{{e^x}}}{{1 + {e^{2x}}}}dx = }$
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Q4 Single Correct Medium

$\displaystyle \int e^{\frac{-x}{2}}\frac{\sqrt{1-\sin x}}{1+\cos x}dX=$
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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.