Problem 16 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check your result by differentiation. $$ \int d r $$

Step-by-Step Solution

Answer

The indefinite integral of \(d r\) is \( r + C \), and this result has been verified through differentiation.

Step 1: Compute the Indefinite Integral

The indefinite integral of \( d r \) (which is the integral of 1 with respect to \( r \)) is simply \( r \). In mathematical form: \[ \int d r = r + C \]Here, \( C \) is the constant of integration which comes up because indefinite integrals represent a family of functions. You will get the same derivative from the function if you differentiate any member of this family.

Step 2: Check the Result by Differentiation

After obtaining the antiderivative, you can verify your result by differentiation. We differentiate \( r + C \) with respect to \( r \), we get \( \frac{d}{dr} (r + C) \).Because both \( r \) and \( C \) are constants, when taking the derivative, you will be left with \( 1 \). This shows that your answer is correct.