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English
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Power SubstitutionDescription A method of integration that aims to replace the nth roots in a function, which are difficult to integrate, with integer powers, which can be easily integrated. For example, if \(\sqrt[4]{x}\) is in the function, we will let \(x={u}^{4}\). |
Examples \[\int \sqrt{5+\sqrt{x}} \, dx\] 1 Use Power Substitution. Let \(u=\sqrt{5+\sqrt{x}}\), \(x={u}^{4}-10{u}^{2}+25\), and \(dx=4{u}^{3}-20u \, du\) 2 Expand. \[\int 4{u}^{4}-20{u}^{2} \, du\] 3 Use Power Rule: \(\int {x}^{n} \, dx=\frac{{x}^{n+1}}{n+1}+C\). \[\frac{4{u}^{5}}{5}-\frac{20{u}^{3}}{3}\] 4 Substitute \(u=\sqrt{5+\sqrt{x}}\) back into the original integral. \[\frac{4{\sqrt{5+\sqrt{x}}}^{5}}{5}-\frac{20{\sqrt{5+\sqrt{x}}}^{3}}{3}\] 5 Add constant. \[\frac{4{(5+\sqrt{x})}^{\frac{5}{2}}}{5}-\frac{20{(5+\sqrt{x})}^{\frac{3}{2}}}{3}+C\] Done  |
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