Question

In the following problems, name the method by identifying whether the integral can be evaluated using substitution, integration by parts, or partial fractions.

Answer 1

Integration by parts or using partial fractions is overkill. Both can be done with simple substitutions.

For the first integral, take y = x + 1 and dy = dx.

`\displaystyle \int \frac2{(x+1)^2} \, dx = \int \frac2{y^2} \, dy = -\frac2y + C \\\\ = -\frac2{x+1} + C`

For the second integral, use the same substitution.

`\displaystyle \int (x-1)/((x+1)^2) \, dx = \int ((y-1)-1)/(y^2) \, dy = \int \left(\frac1y - \frac2{y^2}\right) \, dy = \ln|y| + \frac2y + C \\\\ = \ln|x+1| + \frac2{x+1} + C`