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Section Change of Variable Method

The integrand can often be rewritten so that it is easier to see how one of our derivative theorems can be applied to find the antiderivative of the integrand. This method is called either a change of variable or substitution because you replace some part of the integrand with a new function.

Example 5.1.

Consider \(\dsp \int \frac{x}{\sqrt{x+5}} \; dx.\) If we let \(u(x) = \sqrt{x+5}\text{,}\) then \(\dsp u'(x) = \frac{1}{2\sqrt{x+5}}\) so

\begin{equation*} \int \frac{x}{\sqrt{x+5}} \; dx = 2\int (u^2(x)-5)u'(x) \; dx = 2\int u^2(x)u'(x) - 5u'(x) \; dx = \frac{2}{3} u^3(x) - 10u(x) = \dots. \end{equation*}
Problem 5.2.

Evaluate the indefinite integral, \(\dsp \int x^2(2x^3 + 10)^{15} \; dx,\) using the following steps.

  1. Let \(u(x)=2x^3+10\) and compute \(u'.\)

  2. Show that \(\dsp \int x^2(2x^3 + 10)^{15} \; dx = \frac{1}{6} \int ( u(x) )^{15} u'(x) \; dx\text{.}\)

  3. Evaluate this indefinite integral and then replace \(u(x)\) with \(2x^3+10\) in your answer to eliminate the function \(u\) from your answer.

  4. Verify your answer by taking the derivative.

Problem 5.3.

Evaluate each of the following integrals.

  1. \(\dsp \int (2x+7) \root 3 \of {x^2 + 7x +3} \; dx\)

  2. Three parts to this one!

    1. \(\dsp \int \sqrt{2 - x} \; dx\)

    2. \(\dsp \int x \sqrt{2 -x} \; dx\)

    3. \(\dsp \int x^2 \sqrt{2 - x} \; dx\)

  3. \(\dsp \int \frac{{(\ln x)}^4}{x} \; dx\)

  4. \(\dsp \int {5x^2}{4^{x^3}} \; dx\)

  5. \(\dsp \int_0 ^\frac{\pi}{8} \sin^5(2x) \cos(2x) \; dx\)

  6. Three parts to this one!

    1. \(\dsp \int\frac{x}{(x^2+2)^2} \; dx\)

    2. \(\dsp \int\frac{x}{\sqrt[4]{x^2+2}} \; dx\)

    3. \(\dsp \int \frac{x}{x^2+2} \; dx\)

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