CBSE Class 12 · 2016 · Indefinite IntegralsLong Answer

Find :

\int \frac{( 2 x - 5 ) e ^{2 x}}{( 2 x - 3 ) ^{3}} d x

Answer: $\frac{e ^{2 x}}{2 ( 2 x - 3 ) ^{2}} + C$

Solution

Put

t = 2 x - 3

, so

2 x = t + 3

2 d x = d t

, and

e^{2 x} = e^{t + 3} = e^{3} e^{t}

. Also

2 x - 5 = t - 2

. Then

\int \frac{( 2 x - 5 ) e ^{2 x}}{( 2 x - 3 ) ^{3}} d x = \frac{e ^{3}}{2} \int e^{t} \frac{t - 2}{t ^{3}} d t = \frac{e ^{3}}{2} \int e^{t} (\frac{1}{t ^{2}} - \frac{2}{t ^{3}}) d t .

With

f (t) = \frac{1}{t ^{2}}

f^{'} (t) = - \frac{2}{t ^{3}}

, so

\int e^{t} {f + f^{'}} d t = e^{t} f (t)

\frac{e ^{3}}{2} \cdot e^{t} \cdot \frac{1}{t ^{2}} + C = \frac{e ^{3} e ^{2 x - 3}}{2 ( 2 x - 3 ) ^{2}} + C .

\frac{e ^{2 x}}{2 ( 2 x - 3 ) ^{2}} + C

CBSE Class 12 Board · 2016 · Subjective