_ ^ 3 x + 3 x 4 x + 5 x ;dx = - Vidyadip

MCQ

$\int_{}^{} {\frac{{3\cos x + 3\sin x}}{{4\sin x + 5\cos x}}\;dx = } $

✓
$\frac{{27}}{{41}}x - \frac{3}{{41}}\log (4\sin x + 5\cos x)$
B
$\frac{{27}}{{41}}x + \frac{3}{{41}}\log (4\sin x + 5\cos x)$
C
$\frac{{27}}{{41}}x - \frac{3}{{41}}\log (4\sin x - 5\cos x)$
D
None of these

✓

Answer

Correct option: A.

$\frac{{27}}{{41}}x - \frac{3}{{41}}\log (4\sin x + 5\cos x)$

a
Answer is $(A) 27 / 41 x-3 / 41 \log (4 \sin x+5 \cos x)+c$

$3 \cos x+3 \sin x=a(4 \sin x+5 \cos x)+b(d / d x(4 \sin x+5 \cos x)$

$3 \cos x+3 \sin x=\cos x(5 a+4 b)+\sin x(4 a-5 b)$

Compare the coefficients of sin $x$ and $\cos x$ on the both sides

$(5 a+4 b)=3,(4 a-5 b)=3$

$a=27 / 41, b=-3 / 41$

$\int \frac{3 \cos x+3 \sin x}{4 \sin x+5 \cos x} d x=\int \frac{27}{41} d x-\frac{3}{41} \int \frac{4 \cos x-5 \sin x}{4 \sin x+5 \cos x} d x$

$\int \frac{3 \cos x+3 \sin x}{4 \sin x+5 \cos x} d x=\frac{27}{41} x-\frac{3}{41} \ln |4 \sin x+5 \cos x|+c$

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