Problem 23 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check the result by differentiation. $$ \int 5 u \sqrt[3]{1-u^{2}} d u $$

Step-by-Step Solution

Answer

The indefinite integral of the given function is \(-15/8 (1 - u^{2})^{4/3}\).

Step 1: Define the Components to Use for a Substitution

A suitable substitution to simplify the radical is setting \(v = 1 - u^{2}\). This changes the derivative to \(d v = -2 u d u\). Use this substitution to rewrite the integral in terms of \(v\).

Step 2: Rewrite the Integral in Terms of the New Variable

Here, solve for \(u d u\) from the \(d v\) expression, you get -1/2 dv = u du. Substitute \(u d u\) and \(1-u^{2}\) (\(v\)) in the original integral: \[-1/2 \int 5 v^{1/3} d v \]Simplifying this gets you \[-5/2 \int v^{1/3} d v\]

Step 3: Compute the Integral

To find the integral of this function, you add 1 to the exponent of \(v\) and divide by the new exponent according to the power rule of integration. So, you get \[-5/2 * (3/4) v^{4/3}= -15/8 v^{4/3}\]

Step 4: Substitute Back to the Original Variable

Substitute \(v\) with \(1 - u^{2}\). So, the integral of the function is: \[-15/8 (1 - u^{2})^{4/3}\]

Step 5: Checking the Result by Differentiating

Differentiate this function with respect to \(u\). This gives:\[\frac{d}{d u}(-15/8 (1 - u^{2})^{4/3}) = 5 u (1 - u^{2})^{1/3}\]which is the original function and verifies that the integral was computed correctly.

Key Concepts

Integration by Substitution

Integration by substitution is a technique used to evaluate integrals that are otherwise difficult to integrate directly. Think of it as the reverse process of the chain rule in differentiation. In our exercise, the integral

\[ \int 5 u \sqrt[3]{1-u^{2}} d u \]

becomes more approachable when we choose a substitution to simplify the integrand. By setting

\(v = 1 - u^{2}\),

we transform the integral into a new variable, making it easier to work with. We calculated

\(d v = -2 u d u\),

and therefore

\(-1/2 dv = u du\).

Then we substitute both

\(u du\)

and

\(1-u^{2}\) (\(v\))

in the original integral to rewrite it in terms of \(v\).This technique is invaluable as it transforms complex integrals into simpler forms that are easier to evaluate.

Power Rule of Integration

The power rule of integration is a fundamental concept that allows us to integrate functions of the form

\(x^n\).

To apply the power rule, we increase the exponent by one and divide by the new exponent. For instance, the indefinite integral of

\(x^n\)

is

\(\frac{x^{n+1}}{n+1} + C\),

where \(C\) is the constant of integration. In the given problem, after applying the integration by substitution, we obtained

\[-5/2 \int v^{1/3} d v\],

which requires the power rule for integration. By adding 1 to the exponent and dividing by the new exponent, the integral evaluates to

\[-15/8 v^{4/3}\].

The power rule simplifies the process of finding integrals, making it an essential tool for solving integration problems.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to one of its variables. For the purpose of checking our work in integration, differentiation acts as our quality control. By taking the derivative of our answer and comparing it to the original function, we can confirm the correctness of our integral. In our exercise example, after finding the indefinite integral

\[-15/8 (1 - u^{2})^{4/3}\],

we differentiate this expression with respect to \(u\) to verify our solution. The process of differentiation should reverse integration, leading us back to the integrand of

\[5 u \sqrt[3]{1-u^{2}}\].

If successful, differentiation not only gives us confidence in our work but also deepens our understanding of how integration and differentiation are interconnected operations.