Problem 37 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral and check your result by differentiation. $$ \int \frac{2 x^{3}+1}{x^{3}} d x $$

Step-by-Step Solution

Answer

The indefinite integral \(\int \frac{2 x^{3}+1}{x^{3}} dx\) is \(2x - \frac{1}{x} + C\), and we have verified this by differentiation.

Step 1: Simplify the integrand

Begin by simplifying the integrand. It is a fractional function, so it can be separated into two parts: \( \int \frac{2 x^{3}}{x^{3}} dx + \int \frac{1}{x^{3}} dx \). After simplifying, the integral can be rewritten as: \( \int 2 dx - \int x^{-2} dx \)

Step 2: Perform the Integration

Now, perform the integrals separately. The integral of 2 with respect to x is \(2x + c\), where c is the constant of integration. And the integral of \(x^{-2}\) is \(-x^{-1} + C\), where C is the constant of integration. Altogether, the integral becomes \(2x - x^{-1} + C\) or, which is the same, \(2x - \frac{1}{x} + C\)

Step 3: Verify the Result

Lastly, differentiate the obtained result to ensure it is correct. The derivative of \(2x - \frac{1}{x} + C\) should be the original expression. Applying the rules of differentiation, the derivative becomes \(2 + \frac{1}{x^{2}}\), which simplifies to \(2 + \frac{1}{x^{3}}\). This matches the original expression, thus our integration result is verified.