Problem 24 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{1}{x^{2 / 3}\left(1+x^{1 / 3}\right)} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\int \frac{1}{x^{2 / 3}(1+x^{1/3})} dx\) is \(3 \ln|1+x^{1/3}|\).

Step 1: Identify the substitution

Identifying that by substituting \(u = x^{1/3}\), \(x\) would be equal to \(u^3\) and \(dx\) will be \(3u^2 du\). This substitution simplifies the integral.

Step 2: Substitute

Substitute the values into the integral: \(\int \frac{3u^2}{u^2(1+u)} du\). This simplifies to \(\int \frac{3}{1+u} du\).

Step 3: Calculate the integral

Calculate the integral \(\int \frac{3}{1+u} du\) using the rule for integrals of \(1/x\), which is \(\ln|u|\). So, the result is \(3 \ln|1+u|\).

Step 4: Substitute back the original variable

Substitute back the original variable. The final answer turns into \(3 \ln|1+x^{1/3}|\).

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving integrals, especially when dealing with complex expressions. The main idea behind substitution is to transform a difficult integral into a simpler one by changing the variable. This can often make the process of integration much more straightforward.

In our example, the original integral was\[\int \frac{1}{x^{2/3}(1+x^{1/3})} \, dx\]By choosing a substitution such as \(u = x^{1/3}\), we simplify the expression significantly. This means that \(x = u^3\) and thus, \(dx = 3u^2 \, du\). After substitution, the integral morphs into a more manageable form:

• The denominator \(x^{2/3}(1 + x^{1/3})\) becomes \(u^2(1 + u)\).
• Substituting \(dx\) gives us an integral entirely in terms of \(u\).

Hence, the integral transforms to \(\int \frac{3u^2}{u^2(1+u)} du\), which further simplifies due to cancellation of \(u^2\), leading to the integral \(\int \frac{3}{1+u} \, du\).

Integral Simplification

Simplifying an integral is a critical step that often makes solving it possible and much easier. It involves canceling out terms or simplifying expressions. In the given problem, after performing the substitution, we ended up with\[\int \frac{3u^2}{u^2(1+u)} \, du\]

Upon examining this integral, we notice that \(u^2\) terms in the numerator and the denominator can be canceled out. This simplification reduces the complexity of the integral to:

• \(\int \frac{3}{1+u} \, du\)

Reducing expressions in integrals is a crucial skill. Easier expressions, once simplified, are often straightforward to integrate, which is what we observe here. Removing the \(u^2\) makes the function less cumbersome, demonstrating why simplification is a vital step.

Logarithmic Integration

Logarithmic integration involves integrating functions of the form \(\int \frac{1}{x} \, dx\). Such integrals have a solution in terms of logarithms. Specifically, the integral \(\int \frac{1}{x} \, dx\) equals \(\ln|x|\) plus a constant of integration.

In our problem, after substitution and simplification, the integral needed to integrate was \(\int \frac{3}{1+u} \, du\). This corresponds to a logarithmic integral because the structure has a \(\frac{1}{something}\) form, where "something" is the expression \(1+u\). The antiderivative here is found using the logarithmic rule and is given by:

• \(3 \ln|1+u|\)

Converting back into the original variable \(x\), the final solution is \(3 \ln|1+x^{1/3}|\), showcasing how logarithmic integration simplifies seemingly tough integrals efficiently.