Question

\[\int\frac{x + 2}{\left( x + 1 \right)^3} \text{ dx }\]

Answer 1

\[\text{ Let }\int\frac{\left( x + 2 \right)}{\left( x + 1 \right)^3}dx\]
\[\text{ Putting  x }+ 1 = t\]
\[ \Rightarrow x = t - 1\]
\[ \Rightarrow dx = dt\]
\[ \therefore I = \int\left( \frac{t - 1 + 2}{t^3} \right)dt\]
\[ = \int\left( \frac{1}{t^2} + \frac{1}{t^3} \right)dt\]
\[ = \int\left( t^{- 2} + t^{- 3} \right)dt\]
\[ = \left[ \frac{t^{- 2 + 1}}{- 2 + 1} + \frac{t^{- 3 + 1}}{- 3 + 1} \right] + C \]
\[ = - \frac{1}{t} - \frac{2}{t^2} + C\]
\[ = - \frac{1}{x + 1} - \frac{1}{2 \left( x + 1 \right)^2} + C .......................\left( \because t = x + 1 \right)\]