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Chapter 4: Problem 35

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

Short Answer

Expert verified
Question: Find the indefinite integral of the function $$\frac{4x^4 - 6x^2}{x}$$ and check your result by differentiating. Answer: The indefinite integral of the function $$\frac{4x^4 - 6x^2}{x}$$ is $$x^4 + 3x^2 + C$$, where C is the constant of integration. We have verified this result by differentiating and obtaining the original function.

Step by step solution

01

Simplify the function

We can simplify the given function by dividing the terms separately: $$\int \frac{4 x^{4}-6 x^{2}}{x} d x = \int \frac{4 x^{4}}{x} d x - \int \frac{6 x^{2}}{x} d x$$ This simplifies as: $$\int 4x^3 dx - \int 6x dx$$
02

Integrate the simplified function

Now that the function is simplified, we can integrate each term separately: $$\int 4x^3 dx = 4\int x^3 dx$$ $$\int 6x dx = 6\int x dx$$
03

Apply the Power Rule

The power rule states that for any real number n: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ Applying the power rule to each term, we get: $$4 \int x^3 dx = 4 \left(\frac{x^{3+1}}{3+1} + C_1\right) = x^4 + C_1$$ $$6 \int x dx = 6 \left(\frac{x^{1+1}}{1+1} + C_2\right) = 3x^2 + C_2$$
04

Combine the integration results

Now, we can add the results of the integrations together and combine the constants to get the general solution (indefinite integral): $$\int \frac{4 x^{4} - 6 x^{2}}{x} dx = x^4 + 3x^2 + C$$ where $$C = C_1 + C_2$$ is the constant of integration.
05

Check the solution by differentiating

We'll differentiate the indefinite integral to check if we obtain the initial function back (divide each term by x). To differentiate the function, we use the power rule of differentiation: $$\frac{d}{dx}(x^4) = 4x^3$$ $$\frac{d}{dx}(3x^2) = 6x$$ Then, the derivative of the indefinite integral is: $$\frac{d}{dx}(x^4 + 3x^2 + C) = 4x^3 + 6x$$ Now, we need to divide each term by x to check if we get the initial function back: $$\frac{4x^3 + 6x}{x} = \frac{4 x^{4} - 6 x^2}{x}$$ The result is the same as the initial function, so our indefinite integral is correct.

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