Mathematics

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

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

Realted Questions

Q1 Subjective Hard
Integrate the function    $\cfrac {5x-2}{1+2x+3x^2}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\displaystyle \int_{0}^{\pi /3}\frac{\cos \theta }{3+4\sin \theta }d\theta =\lambda \log \frac{3+2\sqrt{3}}{3}$ then $\displaystyle \lambda$ equals
• A. $\dfrac{1}{2}$
• B. $\dfrac{1}{3}$
• C. $\dfrac{1}{8}$
• D. $\dfrac{1}{4}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Solve :-
$I=4\displaystyle \int_0^{\pi/2}\sin^2{x} \ dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \int\frac{1}{\sqrt{\cos^{4}x-\cos^{2}x\sin^{2}x}}dx$ is equal to
• A. $\sinh^{-1} (\cos x)+c$
• B. $-\sin^{-1} (\tan x)+c$
• C. $\cosh^{-1} (\sin x)+c$
• D. $-\cos^{-1} (\tan x)+c$

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.

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