Solve -sin x-asin x-bdx

The correct option is C I = ( x + b)cos( a b) + sin( a b)log| sin( x + b)| + c ∫_^sin( x + a)sin( x + b)dx putting x+b=t ⇒ x = t b ...

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Standard XII

Mathematics

Functions

Solve ∫sin ...

Question

Solve $\int \frac{sin (x + a)}{sin (x + b)} d x$

I = (x + b) cos (a + b) + sin (a - b) log | sin (x + b) | + c

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I = (x + b) cos (a - b) + sin (a - b) log | sin (x + b) | + c

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I = (x + b) cos (a - b) + sin (a + b) log | sin (x + b) | + c

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Solution

The correct option is C $I = (x + b) cos (a - b) + sin (a - b) log | sin (x + b) | + c$
$\int \frac{sin (x + a)}{sin (x + b)} d x$

putting $x + b = t$ $\Rightarrow x = t - b$
and $d x = d t$
$= \int \frac{sin (t - b + a)}{sin t} d t$

$= \int \frac{sin (t + (a - b))}{sin t} d t$

$= \int \frac{sin t cos (a - b) + cos t sin (a - b)}{sin t} d t$

$= \int cos (a - b) d t + \int sin (a - b) cot t d t$

$= t cos (a - b) + sin (a - b) log | sin t | + c$

$= (x + b) cos (a - b) + sin (a - b) log | sin (x + b) | + c$

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