Chapter 7.2, Problem 25E

To determine

To calculate:

To evaluate the given integral ${\displaystyle \int \frac{dx}{{\left(1-x\right)}^2}}$ by substitution method

Answer to Problem 25E

The solution is $\frac{1}{1-x}+c$

Explanation of Solution

Given information:

The given integral is ${\displaystyle \int \frac{dx}{{\left(1-x\right)}^2}}$ .

Consider the integral ${\displaystyle \int \frac{dx}{{\left(1-x\right)}^2}}$ ,

Let $u=1-x$

Differentiating both sides with respect to x,

  $\frac{du}{dx}=0-1\Rightarrow -1$

Solving this for $dx$ gives,

  $\begin{array}{l}\frac{du}{dx}=-1\\ du=-dx\\ dx=-du\end{array}$

Substitute $u=1-x$ and $dx=-du$ into the integral ${\displaystyle \int \frac{dx}{{\left(1-x\right)}^2}}$ ,

  ${\displaystyle \int \frac{dx}{{\left(1-x\right)}^2}}={\displaystyle \int \frac{-du}{u^2}}$

  $=-{\displaystyle \int u^{-2}du}$

On integrating we get,

  $\begin{array}{l}=-\left(\frac{u^{-1}}{-1}\right)+c\\ =u^{-1}+c\\ =\frac{1}{u}+c\end{array}$

Now substituting back for u,

  $=\frac{1}{1-x}+c$

Therefore, the solution is $\frac{1}{1-x}+c$