Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{d x}{(1-x)^{2}}$$

Short answer

The integral of \(\int \frac{d x}{(1-x)^{2}}\) results in \(1/(1-x)\).

Step-by-step solution

Choose the substitution

Decide on a substitution that will simplify the integral. Here, choosing \(u = 1 - x\) will allow us to convert the integral into a simpler form. Whenever we have a composite function, especially in the denominator, it is a good sign that substitution might work.

Change the differential

Now, find the derivative of \(u = 1 - x\) with respect to \(x\), which results in \(du = -dx\). Let's multiply both sides by -1 to get \(-du = dx\).

Substitute into the integral

Change all \(x\)'s to \(u\)'s in the integral, including \(dx\). Accordingly, our integral will be \(\int \frac{-du}{u^{2}}\). We can treat the negative sign as a constant and move it outside the integral thus making it \(-\int \frac{du}{u^{2}}\).

Evaluate the integral

The integral of \(-\int \frac{du}{u^{2}}\) is a standard form of integral. The integral of \(1/u^2\) is \(-1/u\). Therefore, our integral is \(-(-1/u) = 1/u\).

Reverse the substitution

Substitute \(u\) back in by replacing it with the original \(x\) term, that is, \(u = 1 - x\). The result is \(1/(1-x)\).