Mathematics

Single Correct Medium Published on 17th 09, 2020
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Realted Questions

Q1 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. 

$$\int u(x)\, v(x)dx\, =\, u(x)\, v_{1}(x)\, -\, u^{}(x)v_{2}(x)\, +\, u^{}(x)\, v_{3}(x)\, -\, .\, +\, (-1)^{n\, -\, 1}u^{n\, -\, 1}(x)v_{n}(x)\, -\, (-1)^{n\, -\, 1}$$ $$\int\, u^{n}(x)v_{n}(x)\, dx$$ where $$v_{1}(x)\, =\, \int v(x)dx,\, v_{2}(x)\, =\, \int v_{1}(x)\, dx\, ..\, v_{n}(x)\, =\, \int v_{n\, -\, 1}(x) dx$$

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 $$\int P_{n}(x)\, Q(x)\, dx$$, where $$P_{n}(x)$$, 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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Q2 Multiple Correct Hard
Let $$f(x) = 7\tan^8x+7\tan^6x-3\tan^4x-3\tan^2x$$ for all $$x\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$$, then the correct expression(s) is (are)
  • A. $$\displaystyle \int_0^{\dfrac{\pi}{4}}xf(x)dx = \dfrac{1}{6}$$
  • B. $$\displaystyle \int_0^{\dfrac{\pi}{4}}xf(x)dx =1$$
  • C. $$\displaystyle \int_0^{\dfrac{\pi}{4}}xf(x)dx = \dfrac{1}{12}$$
  • D. $$\displaystyle \int_0^{\dfrac{\pi}{4}}f(x)dx = 0$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q3 Single Correct Hard
 If $$ b > a$$ and $$ \displaystyle I = \int_{a}^{b}\frac{dx}{\sqrt{ (x-a)(b-x)}}$$ then $$I$$ equals
  • A. $$ \pi/2$$
  • B. $$ 3\pi/2$$
  • C. $$ 2\pi$$
  • D. $$ \pi$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q4 Single Correct Hard
If $$\displaystyle A=\int_{0}^{\pi} \dfrac{cos x}{(x+2)^2} \: dx$$, then $$\displaystyle \int_{0}^{\dfrac{\pi}{2}} \dfrac{\sin 2x}{(x+1)} \: dx$$ is equal to
  • A. $$\dfrac{1}{\pi+2}-A$$
  • B. $${1}+\dfrac{1}{\pi+2}-A$$
  • C. $$A-\dfrac{1}{2}-\dfrac{1}{\pi+2}$$
  • D. $$\dfrac{1}{2}+\dfrac{1}{\pi+2}-A$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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Q5 Single Correct Medium
The value of $$\displaystyle \int_{\sqrt{log2}}^{\sqrt{log3}}\dfrac{xsinx^2}{sinx^2+sin(log6-x^2)}dx$$ is
  • A. $$\dfrac{1}{2}log\dfrac{3}{2}$$
  • B. $$log\dfrac{3}{2}$$
  • C. $$\dfrac{1}{6}log\dfrac{3}{2}$$
  • D. $$\dfrac{1}{4}log\dfrac{3}{2}$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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