Mathematics

# Primitive of $\frac{1}{4\sqrt{x}+x}$ is equal to

$2log|4+\sqrt{x}|+c$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Subjective Hard
Find the value of $\displaystyle\int _{ 0 }^{ 2\pi }{ \sin ^{ 2 }{ x } \cdot \cos ^{ 4 }{ x } dx }$.

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \dfrac { d x } { \sin ^ { 2 } x \cos ^ { 2 } x }$ equals-
• A. $\tan x + \cot x + c$
• B. $\tan x \cot x + c$
• C. $\tan x - \cot 2 x + c$
• D. $\tan x - \cot x + C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The value of $\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}\:\dfrac{dx}{e^{sin\:x}+1}dx$ is equal to
• A. $0$
• B. $1$
• C. $-\dfrac{\pi}{2}$
• D. $\dfrac{\pi}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
If $\mathrm{a}>\mathrm{b}$ then $\displaystyle \int_{0}^{\pi}\frac{\mathrm{d}\mathrm{x}}{\mathrm{a}+\mathrm{b}\sin \mathrm{x}}=$
• A. $\displaystyle \frac{1}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\cot^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$
• B. $\displaystyle \frac{1}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\tan^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$
• C. $\displaystyle \frac{2}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\tan^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$
• D. $\displaystyle \frac{2}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\cot^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$

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.