If frac{sin^2 x + 1}{2sin^2 x - 5sin x + 3} = | vedclass

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Question

(A) Let $u = \sin x$. The expression becomes $\frac{u^2 + 1}{2u^2 - 5u + 3}$.Factor the denominator: $2u^2 - 5u + 3 = (2u - 3)(u - 1)$.Perform polynom

Vedclass · Accepted Answer

$A = \frac{13}{2}$

Vedclass · Answer

(A) Let $u = \sin x$. The expression becomes $\frac{u^2 + 1}{2u^2 - 5u + 3}$.Factor the denominator: $2u^2 - 5u + 3 = (2u - 3)(u - 1)$.Perform polynomial division: $\frac{u^2 + 1}{2u^2 - 5u + 3} = \frac{1}{2} + \frac{\frac{5}{2}u - \frac{1}{2}}{2u^2 - 5u + 3}$.Now,decompose $\frac{\frac{5}{2}u - \frac{1}{2}}{(2u - 3)(u - 1)} = \frac{A}{2u - 3} + \frac{B}{u - 1}$.Using partial fractions: $\frac{5u - 1}{2(2u - 3)(u - 1)} = \frac{A}{2u - 3} + \frac{B}{u - 1}$.For $u = 1$: $B = \frac{5(1) - 1}{2(2(1) - 3)} = \frac{4}{-2} = -2$.For $u = \frac{3}{2}$: $A = \frac{5(\frac{3}{2}) - 1}{2(\frac{3}{2} - 1)} = \frac{\frac{13}{2}}{1} = \frac{13}{2}$.The constant term $C = \frac{1}{2}$.Checking the options: $A = \frac{13}{2}$ is correct.

If $\frac{\sin^2 x + 1}{2\sin^2 x - 5\sin x + 3} = \frac{A}{2\sin x - 3} + \frac{B}{\sin x - 1} + C$,then:

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