# Dxab∫dxsin(x-a)sin(x-b) is equal to ______. - Mathematics

MCQ
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int "dx"/(sin(x - "a")sin(x - "b")) is equal to ______.

#### Options

• sin("b" - "a") log|(sin(x - "b"))/(sin(x - "a"))| + "C"

• "cosec"("b" - "a") log|(sin(x - "a"))/(sin(x - "b"))| + "C"

• "cosec"("b" - "a") log|(sin(x - "b"))/(sin(x - "a"))| + "C"

• sin("b" - "a")log|(sin("x" - "a"))/(sin(x - "b"))| + "C"

#### Solution

int "dx"/(sin(x - "a")sin(x - "b")) is equal to "cosec"("b" - "a") log|(sin(x - "b"))/(sin(x - "a"))| + "C".

Explanation:

Let I = int "dx"/(sin(x - "a")sin(x - "b"))

Multiplying and dividing by sin(b – a) we get,

I = 1/(sin("b" - "a")) int (sin("b" - "a"))/(sin(x - "a") * sin(x - "b")) "d"x

= 1/(sin("b" - "a")) int (sin(x + "b" - x - "a"))/(sin(x - "a") * sin(x - "b")) "d"x

= 1/(sin("b" - "a")) int (sin[(x - "a") - (x - "b")])/(sin(x - "a") * sin(x - "b")) "d"x

= 1/(sin("b" - "a")) int (sin(x - "a") cos(x - "b") - cos(x - "a") sin(x - "b"))/(sin(x - "a") * sin(x - "b")) "d"x

= 1/(sin("b" - "a")) int (sin(x - "a") * cos(x - "b"))/(sin(x - "a")*sin(x - "b")) - (cos(x - "a")*sin(x - "b"))/(sin(x - "a") * sin(x - "b")) "d"x

= 1/(sin("b" - "a")) int [(cos(x - "b"))/(sin(x - "b")) - (cos(x - "a"))/(sin(x - "a"))]"d"x

= 1/(sin("b" - "a")) int [cot(x - "b") - cot(x - "a")]"d"x

= 1/(sin("b" - "a")) [log sin(x - "b") - logsin(x - "a")] + "C"

= 1/(sin("b" - "a")) * log|(sin(x - "b"))/(sin(x - "a"))| + "C"

I = "cosec"("b" - "a") log|(sin(x - "b"))/(sin(x - "a"))| + "C".

Is there an error in this question or solution?
Chapter 7: Integrals - Exercise [Page 167]

#### APPEARS IN

NCERT Exemplar Mathematics Class 12
Chapter 7 Integrals
Exercise | Q 49 | Page 167

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