∫$$3sin$$⁡$$x+2cos$$⁡$$x3cos$$⁡$$x+2sin$$⁡xdx is equal to

Question

∫$$3sin$$⁡$$x+2cos$$⁡$$x3cos$$⁡$$x+2sin$$⁡xdx is equal to

Infinity Learn · Accepted Answer

Let $$I=$$∫$$3sin$$⁡$$x+2cos$$⁡$$x3cos$$⁡$$x+2sin$$⁡$$xdx3sin$$⁡$$x+2cos$$⁡$$x=Addx(3cos$$⁡$$x+2sin$$⁡$$x)+B(3cos$$⁡$$x+2sin$$⁡$$x)$$⇒$$3sin$$⁡$$x+2cos$$⁡$$x=A($$−$$3sin$$⁡$$x+2cos$$⁡$$x)+B(3cos$$⁡$$x+2sin$$⁡$$x)On$$ comparing the coefficients of $$sin$$ x and $$cos$$ x of both sides, we get−$$3A+2B=3$$ and $$2A+3B=2$$⇒  $$B=1213$$ and $$A=$$−$$513$$∴ $$I=$$∫$$B($$−$$3sin$$⁡$$x+2cos$$⁡$$x)+A(3cos$$⁡$$x+2sin$$⁡$$x)3cos$$⁡$$x+2sin$$⁡xdx⇒ $$I=A$$∫$$1dx+B$$∫−$$3sin$$⁡$$x+2cos$$⁡$$x3cos$$⁡$$x+2sin$$⁡$$xdx=Ax+B$$∫dtt, where $$t=3cos$$⁡$$x+2sin$$⁡$$x=Ax+Bln$$⁡|t|$$+C=$$−$$513x+1213ln$$⁡|$$3cos$$⁡$$x+2sin$$⁡x|$$+C$$

$\int \frac{3 \sin x + 2 \cos x}{3 \cos x + 2 \sin x} dx$ is equal to

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∫3sin⁡x+2cos⁡x3cos⁡x+2sin⁡xdx is equal to

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$\int \frac{3 \sin x + 2 \cos x}{3 \cos x + 2 \sin x} dx$ is equal to