∫3xCos5x dx==∫elog 3x cos5x dx =∫exlog 3 cos5x dx ∵∫eaxcosbx dx=eaxa2+b2acosbx+b sinbx =elog 3xlog32+25log 3cos5x+5 sin 5x+C =3xlog 32+25log 3cos5x+5 sin 5x+C

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$\int 3^{x} C o s 5 x d x =$

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$\frac{3^{x}}{{(\log 3)}^{2} + 25} [(\log 3) . C o s 5 x - 5 S i n x] + c$

$\frac{3^{x}}{{(\log 3)}^{2} + 25} [(\log 3) . C o s 5 x + 5 S i n 5 x] + c$

$\frac{3^{x}}{{(\log 3)}^{2} + 25} [5 C o s 5 x - (\log 3) . S i n 5 x] + c$

$\frac{3^{x}}{{(\log 3)}^{2} + 25} [5 C o s 5 x + (\log 3) . S i n 5 x] + c$

answer is B.

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Detailed Solution

$\int 3^{x} C o s 5 x d x =$

$= \int e^{\log 3^{x}} cos5x d x = \int e^{x (\log 3)} cos5x d x (∵ \int e^{a x} \cos b x d x = \frac{e^{a x}}{a^{2} + b^{2}} [a \cos b x + b \sin b x]) = \frac{e^{(\log 3) x}}{{(\log 3)}^{2} + 25} [(\log 3) \cos 5 x + 5 \sin 5 x] + C = \frac{3^{x}}{{(\log 3)}^{2} + 25} [(\log 3) \cos 5 x + 5 \sin 5 x] + C$