Problem 34 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int \sqrt[5]{r^{2}} d r$$

Step-by-Step Solution

Answer

Question: Determine the indefinite integral of the following expression and check the result by differentiation: $$\int \sqrt[5]{r^{2}} d r$$ Answer: The indefinite integral of the given expression is $$\frac{5}{7} r^{\frac{7}{5}} + C$$, and by differentiating, we were able to verify that our integration is correct.

Step 1: Rewrite in Exponent Notation

Rewrite the given integral using exponent notation. We can rewrite the nth root as an exponent by showing it as a fraction, 1/n: $$\int \sqrt[5]{r^{2}} d r = \int (r^{2})^{\frac{1}{5}} d r $$

Step 2: Evaluate Using Power Rule

Evaluate the indefinite integral using the power rule for integration, which states that for any real number n other than -1: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ In our problem, we have n = 2/5, so we can apply the power rule as follows: $$\int (r^{\frac{2}{5}}) d r = \frac{r^{\frac{2}{5} + 1}}{\frac{2}{5} + 1} + C$$

Step 3: Simplify the Expression

Add the exponent fractions: $$\frac{2}{5} + 1 = \frac{2}{5} + \frac{5}{5} = \frac{7}{5}$$ Also simplify the denominator: $$\int (r^{\frac{2}{5}}) d r = \frac{r^{\frac{7}{5}}}{\frac{7}{5}} + C = \frac{5}{7} r^{\frac{7}{5}} + C$$

Step 4: Check by Differentiation

To verify that we have integrated correctly, differentiate our result and check that we get the original integrand. Use the power rule for differentiation, which states that for any real number n: $$\frac{d}{dx} (x^n) = nx^{n-1}$$ Differentiate the expression we obtained in step 3: $$\frac{d}{dr} (\frac{5}{7} r^{\frac{7}{5}} + C) = \frac{7}{5} \cdot \frac{5}{7} r^{\frac{7}{5} - 1} = r^{\frac{2}{5}}$$ Since we got the original integrand, our integration is correct. The final solution is: $$\int \sqrt[5]{r^{2}} d r = \frac{5}{7} r^{\frac{7}{5}} + C$$

Key Concepts

Integration Techniques

Integration techniques allow us to find the integral of functions, which is crucial in calculus. One key technique is transforming complicated functions into simpler forms using exponent notation. In the given exercise, the original expression \( \sqrt[5]{r^2} \) represents the fifth root of \( r^2 \). To make it easier to integrate, we rewrite it as \( (r^2)^{1/5} \). This approach helps to apply straightforward rules like the Power Rule for Integration effectively.

The goal is to transform the integrand into a familiar algebraic form, which often involves using laws of exponents. This simplification is fundamental in solving more complex integrals. It enables us to shift from a form that's challenging to work with into one where standard integration techniques can be applied. Practicing rewriting functions in exponent notation is a fundamental skill in mastering integration techniques.

Power Rule

The power rule is a basic and essential formula in integration. It states that the integral of \( x^n \) with respect to \( x \) is \( \frac{x^{n+1}}{n+1} + C \), provided \( n eq -1 \). In our exercise, we applied this rule to integrate \( (r^{2/5}) dr \) by identifying \( n \) as \( \frac{2}{5} \).

Applying the power rule involves two crucial steps:

• Incrementing the original exponent by one. Here, \( \frac{2}{5} + 1 = \frac{7}{5} \).
• Dividing by the new exponent value. Thus, the integral became \( \frac{r^{7/5}}{7/5} \).

Simplifying \( \frac{1}{7/5} \) to obtain \( \frac{5}{7} \) forms the final expression, \( \frac{5}{7} r^{7/5} + C \). This rule is especially handy for polynomial-like functions or transformed expressions like our example. By mastering the power rule, one can efficiently compute integrals without reverting to more complex integration techniques unless necessary.

Differentiation

Differentiation is used to verify the correctness of an integration result. When you differentiate an integral you've computed, you should arrive back at the original function (or integrand). The process essentially serves as a double-check mechanism. In the exercise, we differentiated \( \frac{5}{7} r^{7/5} + C \) to confirm it simplified back to \( r^{2/5} \), the original integrand.

The power rule in differentiation is pivotal here: the derivative of \( x^n \) is \( nx^{n-1} \). For \( \frac{5}{7} r^{7/5} \), the differentiation steps included:

• Multiplying by the exponent \( n \), which is \( \frac{7}{5} \).
• Subtracting one from the exponent to result in \( r^{(7/5) - 1} = r^{2/5} \).

These calculations take you back to the function you initially integrated. Differentiating confirms whether your integration process was done accurately. Knowing how to differentiate and the purpose of doing it after integration is central to solidifying your understanding of calculus principles.