Chapter 16, Problem 1RE

Determine whether each of the following statements is true or false, and explain why.

1. 1. Integration by parts should be used to evaluate ${\displaystyle {\int }_0^1\frac{x^2}{x^3+1}}dx$.

To determine

Whether the given statement is true or false.

Answer to Problem 1RE

The statement is false.

Explanation of Solution

The given statement is “integration by parts is used to evaluate the integral ${\displaystyle \underset{0}{\overset{1}{\int }}\frac{x^2}{x^3+1}}dx$.”

Evaluate above integral through the substitution method as follows,

$\begin{array}{c}{\displaystyle \underset{0}{\overset{1}{\int }}\frac{x^2}{x^3+1}}dx=\frac{1}{3}{\displaystyle \underset{0}{\overset{1}{\int }}\frac{1}{x^3+1}}d\left(x^3+1\right)\\ =\frac{1}{3}{\left[\ln \left|x^3+1\right|\right]}_0^1\\ =\frac{1}{3}\left[\ln \left|1^3+1\right|-\ln \left|0^3+1\right|\right]\\ =\frac{1}{3}\left[\ln 2-\ln 1\right]\end{array}$

On further simplification,

$\begin{array}{c}{\displaystyle \underset{0}{\overset{1}{\int }}\frac{x^2}{x^3+1}}dx=\frac{1}{3}\ln 2-\frac{1}{3}\left(0\right)\\ =\frac{\ln 2}{3}\end{array}$

From the above calculation, it is clear that the integral is evaluated by substitution method instead of integration of by parts.

Thus, the statement is false.