If -sin x-sin x- d x-p x-q log -sin x-C- Then the value of pq is

The correct option is A 12sin 2α∫sin xsin (x α)dx =∫sin (x α+α)sin (x α)dx =∫sin (x α)cosαsin (x α)dx+∫cos (x α)sinαsin(x α)dx =xcosα+sinαln |sin (x α)| +C ...

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Byju's Answer

Standard XII

Mathematics

Theorems on Integration

If ∫sin x/sin...

Question

If $\int \frac{sin x}{sin (x - α)} d x = p x - q log | sin (x - α) | + C$ , Then the value of $p q$ is

- \frac{1}{2} sin 2 α

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sin 2 α

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\frac{1}{2} sin 2 α

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- sin 2 α

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Solution

The correct option is A $- \frac{1}{2} sin 2 α$
$\int \frac{sin x}{sin (x - α)} d x = \int \frac{sin (x - α + α)}{sin (x - α)} d x$
$= \int \frac{sin (x - α) cos α}{sin (x - α)} d x + \int \frac{cos (x - α) sin α}{sin (x - α)} d x$
$= x cos α + sin α ln | sin (x - α) | + C$

$\Rightarrow p = cos α, q = - sin α$
$\Rightarrow p q = - sin α cos α = - \frac{1}{2} sin 2 α$

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If ∫sinxsin(x−α)dx=px−qlog|sin(x−α)|+C, Then the value of pq is

The correct option is A −12sin2α∫sinxsin(x−α)dx=∫sin(x−α+α)sin(x−α)dx =∫sin(x−α)cosαsin(x−α)dx+∫cos(x−α)sinαsin(x−α)dx =xcosα+sinαln|sin(x−α)|+C ⇒p=cosα, q=−sinα ⇒pq=−sinαcosα=−12sin2α

If $\int \frac{sin x}{sin (x - α)} d x = p x - q log | sin (x - α) | + C$ , Then the value of $p q$ is