Mathematics

# Evaluate:$\displaystyle\int_{2}^{3}{\dfrac{\sqrt{x}dx}{\sqrt{5-x}+\sqrt{x}}}$

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

#### Realted Questions

Q1 Single Correct Medium
If $\displaystyle I = \int \frac {e^{3x}}{1 + e^x} dx$, then I equal
• A. $\displaystyle (1/2)(1 + e^x)^2 - (1/3)(1 + e^x) + \log (1 + e^x) + C$
• B. $\displaystyle (1/2)(1 + e^x) (e^x + 3) + \log (1 + e^x) + C$
• C. $\displaystyle (1/2)(1 + e^x)^2 - 2 \log (1 + e^x) + C$
• D. $\displaystyle (1/2)(1 + e^x)(e^x - 3) + \log (1 + e^x) + C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
Slove
$\int \dfrac{3 + 2 cos x}{(2 + 3cos x)^2}dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Hard
The value of the integral $\displaystyle \int \frac {\log (x + 1) - \log x}{x(x + 1)} dx$ is:
• A. $\displaystyle - [(\log (x + 1)^2 - (\log x)^2] + \log (x + 1) \log x + C$
• B. None of these
• C. $\displaystyle -(1/2) (\log (x + 1)^2 - (1/2) (\log x)^2 + \log (x + 1) \log x + C$
• D. $\displaystyle C - (1/2) (\log (1 + 1/x)^2$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate the following integrals:
$\displaystyle \int { \cfrac { 1 }{ { a }^{ 2 }-{ b }^{ 2 }{ x }^{ 2 } } } dx$

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.