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CBSE Class 12 · 2019 · Indefinite IntegralsShort Answer

Find

\int \frac{x - 3}{( x - 1 ) ^{3}} e^{x} d x

Answer: $\frac{e ^{x}}{( x - 1 ) ^{2}} + C$

Solution

Write

x - 3 = (x - 1) - 2

, so

\frac{x - 3}{( x - 1 ) ^{3}} = \frac{1}{( x - 1 ) ^{2}} - \frac{2}{( x - 1 ) ^{3}} .

Take

f (x) = \frac{1}{( x - 1 ) ^{2}}

; then

f^{'} (x) = - \frac{2}{( x - 1 ) ^{3}}

, so the integrand is

e^{x} {f (x) + f^{'} (x)}

and

\int e^{x} {f + f^{'}} d x = e^{x} f (x) + C

:

\frac{e ^{x}}{( x - 1 ) ^{2}} + C

CBSE Class 12 Board · 2019 · Subjective

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