Mathematics

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


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


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Single Correct Medium Published on 17th 09, 2020
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Q1 Subjective Hard
Find the value of $$\displaystyle\int _{ 0 }^{ 2\pi  }{ \sin ^{ 2 }{ x } \cdot \cos ^{ 4 }{ x } dx } $$.

Asked in: Mathematics - Integrals


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Q2 Single Correct Medium
$$\displaystyle \int \dfrac { d x } { \sin ^ { 2 } x \cos ^ { 2 } x }$$ equals-
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Q3 Single Correct Hard
The value of $$\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}\:\dfrac{dx}{e^{sin\:x}+1}dx$$ is equal to 
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  • D. $$\dfrac{\pi}{2}$$

Asked in: Mathematics - Integrals


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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. $$\displaystyle \frac{1}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\cot^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$$
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  • C. $$\displaystyle \frac{2}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}}\tan^{-1}(\frac{\mathrm{b}}{\sqrt{\mathrm{a}^{2}-\mathrm{b}^{2}}})$$
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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.

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