Question

\[\int\frac{e^x \left( x - 4 \right)}{\left( x - 2 \right)^3} \text{ dx }\]

Answer 1

\[\text{ Let I }= \int e^x \left( \frac{x - 4}{\left( x - 2 \right)^3} \right)dx\]

\[ = \int e^x \left[ \frac{x - 2 - 2}{\left( x - 2 \right)^3} \right]dx\]

\[ = \int e^x \left[ \frac{1}{\left( x - 2 \right)^2} - \frac{2}{\left( x - 2 \right)^3} \right]dx\]

\[\text{ Here,  f(x) }= \frac{1}{\left( x - 2 \right)^2}\]

\[ \Rightarrow f'(x) = \frac{- 2}{\left( x - 2 \right)^3}\]

\[\text{ Put e}^x f(x) = t\]

\[ \Rightarrow e^x \frac{1}{\left( x - 2 \right)^2} = t\]

\[\text{ Diff both sides w . r . t x }\]

\[\left[ e^x \frac{1}{\left( x - 2 \right)^2} + e^x \frac{- 2}{\left( x - 2 \right)^3} \right]dx = dt\]

\[ \text{∴  I }= \int dt\]

\[ = t + C\]

\[ = \frac{e^x}{\left( x - 2 \right)^2} + C\]