Mathematics

$$\int { { x }^{ 3 }\tan ^{ -1 }{ x } dx } $$


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Subjective Medium Published on 17th 09, 2020
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Integrate the function    $$\cfrac {5x-2}{1+2x+3x^2}$$

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$$\displaystyle \int_{0}^{\pi /3}\frac{\cos \theta }{3+4\sin \theta }d\theta =\lambda \log \frac{3+2\sqrt{3}}{3}$$ then $$\displaystyle \lambda $$ equals
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Q3 Subjective Medium
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Q4 Single Correct Medium
$$\displaystyle \int\frac{1}{\sqrt{\cos^{4}x-\cos^{2}x\sin^{2}x}}dx$$ is equal to
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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. 

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

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1 Verified Answer | Published on 17th 09, 2020

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