int frac{3 sin x-5 cos x}{7 cos x+2 sin x} , | vedclass

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Question

(A) To solve the integral $I = \int \frac{3 \sin x - 5 \cos x}{2 \sin x + 7 \cos x} \, dx$,we express the numerator as $A(	ext{denominator}) + B(\fra

Vedclass · Accepted Answer

$-\frac{29}{53} x - \frac{31}{53} \log |7 \cos x + 2 \sin x| + c$

Vedclass · Answer

(A) To solve the integral $I = \int \frac{3 \sin x - 5 \cos x}{2 \sin x + 7 \cos x} \, dx$,we express the numerator as $A(	ext{denominator}) + B(\frac{d}{dx}(	ext{denominator}))$.Let $3 \sin x - 5 \cos x = A(2 \sin x + 7 \cos x) + B(2 \cos x - 7 \sin x)$.Equating the coefficients of $\sin x$ and $\cos x$:For $\sin x$: $2A - 7B = 3$For $\cos x$: $7A + 2B = -5$Solving these equations: Multiply the first by $2$ and the second by $7$: $4A - 14B = 6$ and $49A + 14B = -35$.Adding them: $53A = -29 \implies A = -\frac{29}{53}$.Substituting $A$ into $2A - 7B = 3$: $2(-\frac{29}{53}) - 7B = 3 \implies -\frac{58}{53} - 3 = 7B \implies 7B = -\frac{217}{53} \implies B = -\frac{31}{53}$.Thus,$I = \int \frac{A(2 \sin x + 7 \cos x) + B(2 \cos x - 7 \sin x)}{2 \sin x + 7 \cos x} \, dx = A \int 1 \, dx + B \int \frac{2 \cos x - 7 \sin x}{2 \sin x + 7 \cos x} \, dx$.$I = A x + B \log |2 \sin x + 7 \cos x| + c = -\frac{29}{53} x - \frac{31}{53} \log |2 \sin x + 7 \cos x| + c$.

$\int \frac{3 \sin x-5 \cos x}{7 \cos x+2 \sin x} \, dx =$

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