Problem 23 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(3 x^{5}-5 x^{9}\right) d x$$

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the function \(3x^5 - 5x^9\), and check your result by differentiation. Answer: The indefinite integral of the given function is \(\frac{3x^6}{6} - \frac{5x^{10}}{10} + C\), where C is the constant of integration. The derivative of this result is \(3x^5 - 5x^9\), which matches the original function, confirming the correctness of the result.

Step 1: Integrate each term separately

To find the indefinite integral of the given function \((3x^5 - 5x^9)\), we will integrate each term separately, which means finding the antiderivative of both \(3x^5\) and \(-5x^9\). For the antiderivative of each term, we will use the power rule, which states that: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ where \(C\) is the constant of integration.

Step 2: Integrate the term \(3x^5\)

Applying the power rule for the term \(3x^{5}\), we have: $$\int 3x^5 dx = 3\int x^5 dx = 3\cdot \frac{x^{5+1}}{5+1} + C_1$$ $$\int 3x^5 dx = \frac{3x^6}{6} + C_1$$

Step 3: Integrate the term \(-5x^9\)

Applying the power rule for the term \(-5x^9\), we have: $$\int -5x^9 dx = -5\int x^9 dx = -5\cdot \frac{x^{9+1}}{9+1} + C_2$$ $$\int -5x^9 dx = -\frac{5x^{10}}{10} + C_2$$

Step 4: Combine the results

Combining the results from Step 2 and Step 3, we get the indefinite integral of the given function: $$\int (3x^5 - 5x^9)dx = \frac{3x^6}{6} - \frac{5x^{10}}{10} + C$$ Where \(C = C_1 + C_2\) is the constant of integration.

Step 5: Check the result by differentiation

Now we will check our work by differentiating the result we obtained in Step 4: $$\frac{d}{dx}\left(\frac{3x^6}{6} - \frac{5x^{10}}{10} + C\right)$$ Using the power rule for differentiation, we have: $$\frac{d}{dx}\left(\frac{3x^6}{6}\right) = 3x^5$$ $$\frac{d}{dx}\left(-\frac{5x^{10}}{10}\right) = -5x^9$$ $$\frac{d}{dx}(C) = 0$$ So, the derivative of our antiderivative is: $$3x^5 - 5x^9$$ Since the derivative is equal to the original function, we have found the correct indefinite integral.