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
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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. $$\displaystyle (1/2)(1 + e^x)(e^x - 3) + \log (1 + e^x) + C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


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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. $$\displaystyle C - (1/2) (\log (1 + 1/x)^2$$

Asked in: Mathematics - Integrals


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

Asked in: Mathematics - Integrals


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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.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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