If ( ∫ (2 x- x+λ)/( x+ x-2) d x) (=A | x+ x-2|+B x+C). Then the ordered triplet (A, B, λ) is -

Correct answer: B. ( ((3)/(2), (1)/(2),- 1)) aligned d dx (A | x+ x-2|+Bx+C) &= A x- x x+ x-2 +B &= A x-A x+B x+B x-2 B x+ x-2 & 2=A+B or -1=-A+B; λ=-2 B & A=3 / 2, B=1 / 2, λ=-1 aligned )

BITSAT 2009

Exam: BITSAT
Year: 2009
Subject: Mathematics
Chapter: Indefinite Integration

BITSAT 2009 · Mathematics · Indefinite Integration

If ( ∫ (2 x- x+λ)/( x+ x-2) d x) (=A | x+ x-2|+B x+C). Then the...

If (

\int \frac{2 \cos x-\sin x+\lambda}{\cos x+\sin x-2}

d x) (=A \ln |\cos x+\sin x-2|+B x+C). Then the ordered triplet (A, B, \lambda) is -

Options

A( $\left(\frac{1}{2}, \frac{3}{2},-$ 1\right))
B( $\left(\frac{3}{2}, \frac{1}{2},-$ 1\right))
C( $\left(\frac{1}{2},-$ 1, $\frac{3}{2}\right)$ )
D( $\left(\frac{3}{2},-$ 1, $\frac{1}{2}\right)$ )

Correct answer

B. ( $\left(\frac{3}{2}, \frac{1}{2},-$ 1\right))

Step-by-step solution

$\begin{aligned} \frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{A} \ln |\cos \mathrm{x}+\sin \mathrm{x}-2|+\mathrm{Bx}+\mathrm{C})\\ &= \mathrm{A} \frac{\cos \mathrm{x}-\sin \mathrm{x}}{\cos \mathrm{x}+\sin \mathrm{x}-2}+\mathrm{B}\\ &= \frac{\mathrm{A} \cos \mathrm{x}-\mathrm{A} \sin \mathrm{x}+\mathrm{B} \cos \mathrm{x}+\mathrm{B} \sin \mathrm{x}-2 \mathrm{ B}}{\cos \mathrm{x}+\sin \mathrm{x}-2}\\ & \therefore 2=\mathrm{A}+\mathrm{B} \text{ or }-1=-\mathrm{A}+\mathrm{B}; \lambda=-2 \mathrm{ B}\\ & \therefore \mathrm{A}=3 / 2, \mathrm{ B}=1 / 2, \lambda=-1 \end{aligned}

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