Let -0-2 be fixed- If the integral -tan x-tan-tan x-tan-dx-Axcos2-Bxsin2-C- where C is a constant of integration- then the functions Ax and Bx are respectively-

∫tan x+tanαtan x tanαdx ⇒∫sin xcosα+cos xsinαsin xcosα cos xsinαdx =∫sin (x+α)sin (x α)dx =∫sin (x α+2α)sin (x α)dx =∫sin (x α)cos2αsin (x α)dx+∫cos (x α)sin ...

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Standard XIII

Mathematics

Integration by Partial Fractions

Let α∈0,π2 be...

Question

Let $α \in (0, \frac{π}{2})$ be fixed. If the integral $\int \frac{tan x + tan α}{tan x - tan α} d x = A (x) cos 2 α + B (x) sin 2 α + C,$ where $C$ is a constant of integration, then the functions $A (x)$ and $B (x)$ are respectively:

x - α

and

{log}_{e} | sin (x - α) |

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x + α

and

{log}_{e} | sin (x - α) |

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x - α

and

{log}_{e} | cos (x - α) |

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x + α

and

{log}_{e} | sin (x + α) |

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Solution

The correct option is A $x - α$ and ${log}_{e} | sin (x - α) |$
$\int \frac{tan x + tan α}{tan x - tan α} d x$
$\Rightarrow \int \frac{sin x cos α + cos x sin α}{sin x cos α - cos x sin α} d x = \int \frac{sin (x + α)}{sin (x - α)} d x = \int \frac{sin (x - α + 2 α)}{sin (x - α)} d x = \int \frac{sin (x - α) cos 2 α}{sin (x - α)} d x + \int \frac{cos (x - α) sin 2 α}{sin (x - α)} d x = x cos 2 α + sin 2 α log | sin (x - α) | + c$

Here, option $(a)$ and $(b)$ both are correct so it is bonus.

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Similar questions

Q. Let

α \in (0, \frac{π}{2})

be fixed. If the integral

\int \frac{tan x + tan α}{tan x - tan α} d x = A (x) cos 2 α + B (x) sin 2 α + C,

where

C

is a constant of integration, then the functions

A (x)

and

B (x)

are respectively:

Q. If

\int {sin}^{- 1} (\sqrt{\frac{x}{1 + x}}) d x = A (x) {tan}^{- 1} (\sqrt{x}) + B (x) + C,

where

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(A (x), B (x))

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Q. If

\int \frac{\sqrt{x}}{x + 1} d x = A \sqrt{x} + B {tan}^{- 1} \sqrt{x} + C,

then

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is integration constant and

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and

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are fixed constants)

Q. If

\int \frac{d x}{4 - 3 {cos}^{2} x + 5 {sin}^{2} x} = A f (3 tan x) + C,

for a combination of function

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Q. If

\int \frac{3 e^{x} - 5 e^{- x}}{4 e^{x} + 5 e^{- x}} d x = A x + B ln | 4 e^{x} + 5 e^{- x} | + C,

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Let α∈(0,π2) be fixed. If the integral ∫tanx+tanαtanx−tanαdx=A(x)cos2α+B(x)sin2α+C, where C is a constant of integration, then the functions A(x) and B(x) are respectively:

Let $α \in (0, \frac{π}{2})$ be fixed. If the integral $\int \frac{tan x + tan α}{tan x - tan α} d x = A (x) cos 2 α + B (x) sin 2 α + C,$ where $C$ is a constant of integration, then the functions $A (x)$ and $B (x)$ are respectively: