If- I-d x-sinx-3 cos x then I equalsA- None of theseB- 2 log-sin x-sinx-3-CC- 2 log-sinx-3 secx-CD- 2 log-sin x-sinx-3-C

The correct option is C 2log | sin ( x π/3 )sec x |+CI=1/cos ( π/3 )∫cos ( x ( x π/3 ) )/sin ( x π/3 )cos x =1/cosπ/3∫cos(x (x π/3 ) )/sin(x π/3 )cos xdx ...

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Byju's Answer

Standard XII

Mathematics

Standard Formulae - 2

If, I=∫d x/si...

Question

If, $I = \int \frac{d x}{s i n (x - \frac{π}{3}) c o s x}$ then I equals

2 l o g ∣ ∣ s i n x + s i n (x - \frac{π}{3}) ∣ ∣ + C

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2 l o g ∣ ∣ s i n (x - \frac{π}{3}) s e c x ∣ ∣ + C

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2 l o g ∣ ∣ s i n x - s i n (x - \frac{π}{3}) ∣ ∣ + C

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Solution

The correct option is B $2 l o g ∣ ∣ s i n (x - \frac{π}{3}) s e c x ∣ ∣ + C$
$I = \frac{1}{c o s (\frac{π}{3})} \int \frac{c o s (x - (x - \frac{π}{3}))}{s i n (x - \frac{π}{3}) c o s x}$
$= \frac{1}{c o s \frac{π}{3}} \int \frac{c o s (x - (x - \frac{π}{3}))}{s i n (x - \frac{π}{3}) c o s x} d x$
$= \frac{1}{c o s \frac{π}{3}} \int \frac{(c o s x c o s (x - \frac{π}{3}) + s i n x s i n (x - \frac{π}{3}))}{s i n (x - \frac{π}{3}) c o s x} d x$
$= \frac{1}{c o s \frac{π}{3}} \int (c o t (x - \frac{π}{3}) + t a n x) d x$
$= 2 (l o g ∣ ∣ s i n (x - \frac{π}{3}) ∣ ∣ + l o g | s e c x |) + c$

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