Question

Make sure that a formal change of the variable \(\mathrm{t}=\mathrm{x}^{2 / 5}\) leads to the wrong result in the integral \(\int_{-2}^{2} \sqrt[5]{\mathrm{x}^{2}} \mathrm{dx} .\) Find the mistake and explain it.

Short answer

Answer: The substitution \(t = x^{\frac{2}{5}}\) leads to the wrong result because it loses information about the negative values of \(x\). When using this substitution, both \((-2)^{\frac{2}{5}}\) and \((2)^{\frac{2}{5}}\) map to the same value, causing the integral limits to be the same and thus the integral evaluates incorrectly. To evaluate the integral correctly, we should either split it into two parts (positive side and negative side) or find an appropriate substitution that doesn't lose information about the negative side.

Step-by-step solution

Original Integral

First, let's write down the original integral: $$ \int_{-2}^{2} \sqrt[5]{x^2} \,dx. $$

Substitution of variable

Now, we make the substitution \(t = x^{\frac{2}{5}}\), so \(x = t^{\frac{5}{2}}\). Let's evaluate the derivatives of \(x\) with respect to \(t\): $$\frac{dx}{dt} = \frac{5}{2} t^{\frac{3}{2}}.$$ Now, we need to change the integral limits according to the new variable, \(t\): $$t_{-2} = (-2)^{\frac{2}{5}} = 2^{\frac{2}{5}}, \quad t_{2} = (2)^{\frac{2}{5}} = 2^{\frac{2}{5}}.$$

New Integral

Now we can change the integral in terms of \(t\), which gives us: $$ \int_{2^{\frac{2}{5}}}^{2^{\frac{2}{5}}} \sqrt[5]{(t^{\frac{5}{2}})^2} \, \frac{5}{2} t^{\frac{3}{2}} \, dt. $$

Simplification

Now, let's simplify the integral: $$ \frac{5}{2}\int_{2^{\frac{2}{5}}}^{2^{\frac{2}{5}}} t^{\frac{3}{2}} \, t^{\frac{5}{2}} \, dt. $$ Combining the powers of \(t\), we get: $$ \frac{5}{2}\int_{2^{\frac{2}{5}}}^{2^{\frac{2}{5}}} t^4 \, dt. $$

Observations & Explanation of the Mistake

At this point, we can observe that the limits of integration are the same, namely \(2^{\frac{2}{5}}\). This is the mistake in our substitution, because we lose the information about the negative side of the integral when we substitute \(t = x^{\frac{2}{5}}\) since both \((-2)^{\frac{2}{5}}\) and \((2)^{\frac{2}{5}}\) map to the same value. When we substituted \(t = x^{\frac{2}{5}}\), we lost information about the negative values of \(x\). The fifth root of a number squared will always be positive, so the substitution does not account for the negative values of the original integral, creating an incorrect result. To avoid this error while solving this integral, we should either split the original integral into two parts (positive side and negative side), or find an appropriate substitution that doesn't lose information about the negative side.