If 3 x-2 x/4 x+5 x \, dx = A |5 x+4 x|+Bx+c, then A and B are _____

Correct Answer is : A=2241&B=741A=22/41 \& B=7/41A=4122&B=417

Correct Answer is : A=2241&B=741A=22/41 \& B=7/41A=4122&B=417

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AP EAPCET 24-Aug-2021 Shift 2 Paper

Section: Mathematics

Question : 76 of 160

Marks: +1, -0

\int \frac{3\cos x-2\sin x}{4\sin x+5\cos x} \, dx

= A \log|5\cos x+4\sin x|+Bx+c

A

B

are _____

Solution:

We have,

\int \frac{3\cos x-2\sin x}{4\sin x+5\cos x} \, dx

= A(5\cos x+4\sin x)+Bx+c

On differentiating both sides, we get

\frac{3\cos x-2\sin x}{4\sin x+5\cos x}=\frac{A(-5\sin x+4\cos x)}{4\sin x+5\cos x}+B

\Rightarrow \frac{3\cos x-2\sin x}{4\sin x+5\cos x}

=\frac{-5A\sin x+4A\cos x+4B\sin x+5B\cos x}{4\sin x+5\cos x}

\Rightarrow 3\cos x-2\sin x=(4B-5A)\sin x+(5B+4A)\cos x

Equating the coefficient of

\cos x

and

\sin x

, we get

5B+4A=3

and

5A-4B=2

Solving, we get

A=\frac{22}{41}

and

B=\frac{7}{41}

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