{cosec}(x-a) · {cosec} x d x=

Correct Answer is : 1sina⁡log⁡(sin⁡(x−a)⋅cosec⁡x)+c{1}{{sina}} ( (x-a) · {cosec} x)+csina1log(sin(x−a)⋅cosecx)+c, where c is a constant of integration.

-1/ a ( (x-a) x)+c, where c is a constant of integration.

1/ a ( (x-a) x)+c, where c is a constant of integration.

{1}{{sina}} ( (x-a) · {cosec} x)+c, where c is a constant of integration.

{-1}{{sina}} ({cosec}(x-a) · x)+c, where c is a constant of integration.

{cosec}(x-a) · {cosec} x d x=

Correct Answer is : 1sina⁡log⁡(sin⁡(x−a)⋅cosec⁡x)+c{1}{{sina}} ( (x-a) · {cosec} x)+csina1log(sin(x−a)⋅cosecx)+c, where c is a constant of integration.

Test Index

MHTCET 15 May 2024 Shift 2 Paper

Question : 106 of 150

Marks: +1, -0

\int \operatorname{cosec}(x-a) \cdot \operatorname{cosec} x d x=

$\frac{-1}{\sin a} \log (\sin (x-a) \sin x)+c$ , where c is a constant of integration.
$\frac{1}{\sin a} \log (\sin (x-a) \sin x)+c$ , where c is a constant of integration.
$\frac{1}{\operatorname{sina}} \log (\sin (x-a) \cdot \operatorname{cosec} x)+c$ , where c is a constant of integration.
$\frac{-1}{\operatorname{sina}} \log (\operatorname{cosec}(x-a) \cdot \sin x)+c$ , where c is a constant of integration.

Validate

Solution:

Concept:

Use the identity

\sin a = \sin[x-(x-a)]

to split the integrand into a difference of cotangents, which integrates directly.

Explanation:

Let

I = \int \operatorname{cosec}(x-a) \cdot \operatorname{cosec} x \, dx

I = \int \frac{1}{\sin(x-a)\sin x} \, dx

Multiply numerator and denominator by

\sin a

I = \frac{1}{\sin a} \int \frac{\sin[x-(x-a)]}{\sin(x-a)\sin x} \, dx

Since

\sin[x-(x-a)] = \sin x\cos(x-a) - \cos x\sin(x-a)

, we get:

I = \frac{1}{\sin a} \int [\cot(x-a) - \cot x] \, dx

Integrating term by term,

\int \cot(x-a)\, dx = \log|\sin(x-a)|

and

\int \cot x\, dx = \log|\sin x|

Therefore,

I = \frac{1}{\sin a} [\log|\sin(x-a)| - \log|\sin x|] + c

I = \frac{1}{\sin a} \log\left|\frac{\sin(x-a)}{\sin x}\right| + c

I = \frac{1}{\sin a} \log|\sin(x-a)\cdot \operatorname{cosec} x| + c

Answer:

Option C:

\frac{1}{\sin a} \log(\sin(x-a)\cdot \operatorname{cosec} x) + c

, where

c

is the constant of integration.

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