int frac{3cos x + 3sin x}{4sin x + 5cos x} , | vedclass

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Question

(A) To evaluate the integral $I = \int \frac{3\cos x + 3\sin x}{4\sin x + 5\cos x} \, dx$,we express the numerator as $A(	ext{denominator}) + B(\frac

Vedclass · Accepted Answer

$\frac{27}{41}x - \frac{3}{41}\ln |4\sin x + 5\cos x| + C$

Vedclass · Answer

(A) To evaluate the integral $I = \int \frac{3\cos x + 3\sin x}{4\sin x + 5\cos x} \, dx$,we express the numerator as $A(	ext{denominator}) + B(\frac{d}{dx}(	ext{denominator}))$.Let $3\cos x + 3\sin x = A(4\sin x + 5\cos x) + B(4\cos x - 5\sin x)$.Equating the coefficients of $\sin x$ and $\cos x$:For $\sin x$: $4A - 5B = 3$For $\cos x$: $5A + 4B = 3$Solving these simultaneous equations:Multiply the first by $4$ and the second by $5$: $16A - 20B = 12$ and $25A + 20B = 15$.Adding them: $41A = 27 \implies A = \frac{27}{41}$.Substituting $A$ into $5A + 4B = 3$: $5(\frac{27}{41}) + 4B = 3 \implies \frac{135}{41} + 4B = 3 \implies 4B = 3 - \frac{135}{41} = \frac{123 - 135}{41} = -\frac{12}{41} \implies B = -\frac{3}{41}$.Thus,$I = \int \frac{A(4\sin x + 5\cos x) + B(4\cos x - 5\sin x)}{4\sin x + 5\cos x} \, dx = A \int 1 \, dx + B \int \frac{4\cos x - 5\sin x}{4\sin x + 5\cos x} \, dx$.$I = \frac{27}{41}x - \frac{3}{41} \ln |4\sin x + 5\cos x| + C$.

$\int \frac{3\cos x + 3\sin x}{4\sin x + 5\cos x} \, dx = $

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