Problem 35 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{4 x^{4}-6 x^{2}}{x} d x$$

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the function $$\frac{4 x^{4}-6 x^{2}}{x}$$, and check your work by differentiation. Answer: The indefinite integral of the function $$\frac{4 x^{4}-6 x^{2}}{x}$$ is $$x^4 - 3x^2 + C$$.

Step 1: Separate the terms

First, we will separate the given function into two separate terms, which will allow us to integrate each term individually: $$\int \frac{4 x^{4}-6 x^{2}}{x} d x = \int \frac{4 x^{4}}{x} d x - \int \frac{6 x^{2}}{x} d x$$

Step 2: Simplify the terms

Next, we will simplify each term by dividing the numerator by the denominator. Then we will rewrite the integral: $$\int \frac{4 x^{4}}{x} d x - \int \frac{6 x^{2}}{x} d x = \int 4x^3 dx - \int 6x dx$$

Step 3: Integrate each term

Now we will integrate each term. For the first term, we add 1 to the exponent (3 + 1 = 4) and divide by the new exponent. For the second term, we add 1 to the exponent (1 + 1 = 2) and divide by the new exponent. $$\int 4x^3 dx - \int 6x dx = \frac{4}{4}x^4 - \frac{6}{2}x^2 + C$$

Step 4: Simplify the result

We simplify the result, don't forget to add constant C, as this is an indefinite integral: $$\frac{4}{4}x^4 - \frac{6}{2}x^2 + C = x^4 - 3x^2 + C$$

Step 5: Check by differentiation

Finally, we check our work by differentiating our result. If the derivative matches our original function, then we have found the correct integral: $$\frac{d}{d x} (x^4 - 3x^2 + C) = 4x^3 - 6x$$ Which matches our original function, thus confirming the correctness of our integration.