Problem 41 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int 12 x \sqrt[5]{1+6 x^{2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{5}{6}(1+6x^2)^{6/5} + C\).

Step 1: Identify the Integration Method

This integral is a candidate for substitution because the integrand includes a composite function: powers of a binomial expression, \(1 + 6x^2\). By inspecting the integral, we can decide to use substitution where \(u = 1 + 6x^2\).

Step 2: Perform u-Substitution

Let \(u = 1 + 6x^2\), then calculate the differential: \(du = 12x \, dx\). This substitution simplifies the integral to \(\int u^{1/5} \, du\), replacing \(12x \, dx\) with \(du\).

Step 3: Integrate Using the Power Rule

The integral \(\int u^{1/5} \, du\) can be solved using the power rule. Integrate to get \(\frac{u^{6/5}}{6/5}\) or equivalently \(\frac{5}{6}u^{6/5} + C\), where \(C\) is the constant of integration.

Step 4: Substitute Back the Original Variable

Replace \(u\) with \(1+6x^2\) to revert back to the original variable: \(\frac{5}{6}(1+6x^2)^{6/5} + C\).

Step 5: Verify by Differentiation

Differentiate \(\frac{5}{6}(1+6x^2)^{6/5} + C\) with respect to \(x\). Using the chain rule, the derivative is \(\frac{5}{6} \cdot \frac{6}{5} (1+6x^2)^{1/5} \cdot 12x = 12x (1+6x^2)^{1/5}\), confirming that it matches the original integrand.

Key Concepts

u-substitution

One of the most powerful techniques for tackling integrals is u-substitution. It helps transform complex integrals into simpler, more manageable forms. This is particularly useful when dealing with composite functions. The idea is to choose a substitution that simplifies the expression you're integrating. In this exercise, we identify the form of the integrand, which is a binomial expression raised to a power. This signals that u-substitution can handle it effectively.

To start, select a part of the integrand to be the new variable "u". A strategic choice is a component whose derivative is also present in the integrand. Here, setting:

• u = 1 + 6x^2" aligns with this strategy.
• The derivative of "u", denoted as "\(du\)", equals "12x \, dx".

Substituting these values into the integral changes its format, making it much simpler: "\(\int u^{1/5} \, du\)".

This process not only simplifies the integral but also underscores the importance of recognizing patterns within expressions.

power rule

The power rule is an essential integration and differentiation technique used to handle functions of the form \(u^n\). It simplifies finding antiderivatives, enabling you to integrate powers of variables quickly. The rule states that for any power "\(n\)" (except \(n = -1\)), the integral of \(u^n\) with respect to \(u\) is given by:

• \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\), where \(C\) is the constant of integration.
• This provides a direct method to find antiderivatives by increasing the exponent by one.

In our exercise, after performing u-substitution, the integral becomes "\(\int u^{1/5} \, du\)". Using the power rule, integrate to get:

• \(\frac{u^{6/5}}{6/5} + C\).
• Simplified, this gives \(\frac{5}{6} u^{6/5} + C\).

This step significantly reduces the complexity of the problem and aids in further integral evaluations.

differentiation

Differentiation is the process of finding a derivative, which gives the rate at which one quantity changes with respect to another. It is pivotal in verifying integration results. After integrating and substituting back to the original variable in the given example, differentiation confirms the correctness of the solution.

To validate our integrated result, "\(\frac{5}{6}(1+6x^2)^{6/5} + C\)", we need to differentiate it:

• The chain rule helps find the derivative of composite functions.
• Differentiate: \(\frac{5}{6} \cdot \frac{6}{5} (1 + 6x^2)^{1/5} \cdot 12x\).

This simplifies back to the integrand: "\(12x (1 + 6x^2)^{1/5}\)".

Successfully matching the original integrand verifies that the integration and substitution were executed correctly. Differentiation hence acts as a reliable checkpoint in the integration process.