Mathematics

# The value of $\displaystyle \int_{-2\pi}^{5\pi} \cot^{-1}(\tan x) dx$ is equal to:

None of these

##### SOLUTION

Let $\cot^{-1}(\tan x) = \theta$

Then, $\tan x = \cot \theta$

Differentiating,

$\sec^2 x \; dx = -\csc ^2 \theta \; d\theta$

$dx = -\dfrac{\csc ^2 \theta }{\sec^2 x}\; d\theta$

Since

$\tan x = \cot \theta$

$\Rightarrow \pm \sqrt{\sec^2 x - 1} = \cot \theta$

$\sec^2 x - 1 = \cot^2 \theta$

$\sec^2 x = 1 + \cot^2 \theta = \csc^2 \theta$

Hence,

$dx = -\dfrac{\csc ^2 \theta }{\sec^2 x}\; d\theta = -d\theta$

$\therefore \int \cot^{-1}(\tan x) dx= -\int \theta d\theta$

$= -\dfrac{\theta^2}{2} + c$

Since the function $y = \cot^{-1}x$ has the range $(0,\pi)$,

$\cot^{-1}(\tan (-2\pi)) = \cot^{-1}(0) = \dfrac{\pi}{2}$

$\cot^{-1}(\tan (5\pi)) = \cot^{-1}(0) = \dfrac{\pi}{2}$

$\therefore \displaystyle \int_{-2\pi}^{5\pi} \cot^{-1}(\tan x) dx= -\dfrac{\theta^2}{2} \bigg |_{\frac{\pi}{2}}^{\frac{\pi}{2}} = 0$

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Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 114

#### Realted Questions

Q1 Subjective Medium
Evaluate $\displaystyle \int\cfrac{x^{2}}{\sqrt{x^{6}+a^{6}}} \ dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Assertion & Reason Hard
##### ASSERTION

Consider the function $F(x)=\displaystyle \int \frac{x}{(x-1)(x^{2}+1)}dx$

STATEMENT-1 : $F(x)$ is discontinuous at $x=1$

##### REASON

STATEMENT-2 : Integrand of $F(x)$ is discontinuous at $x=1$

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
• B. STATEMENT-1 is True, STATEMENT-2 is False
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Q5 Passage Hard

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