5.5 Substitution  (Page 2/6)

Rewrite the integral as ${\displaystyle \int z{\left(z^2-5\right)}^{1\text{/}2}dz}.$ Let $u=z^2-5$ and $du=2zdz.$ Now we have a problem because $du=2zdz$ and the original expression has only $zdz.$ We have to alter our expression for du or the integral in u will be twice as large as it should be. If we multiply both sides of the du equation by $\frac{1}{2}.$ we can solve this problem. Thus,

Write the integral in terms of u , but pull the $\frac{1}{2}$ outside the integration symbol:

Integrate the expression in u :

We know the derivative of $\text{cos}t$ is $\text{−}\text{sin}t,$ so we set $u=\text{cos}t.$ Then $du=\text{−}\text{sin}tdt.$ Substituting into the integral, we have

Evaluating the integral, we get

Putting the answer back in terms of t , we get

If we let $u=x-1,$ then $du=dx.$ But this does not account for the x in the numerator of the integrand. We need to express x in terms of u . If $u=x-1,$ then $x=u+1.$ Now we can rewrite the integral in terms of u :

Then we integrate in the usual way, replace u with the original expression, and factor and simplify the result. Thus,