Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. $$\int\left(x^{6}-3 x^{2}\right)^{4}\left(x^{5}-x\right) d x$$

Short answer

Question: Find the indefinite integral of the given function: $$\int\left(x^{6}-3 x^{2}\right)^{4}\left(x^{5}-x\right) d x$$ Answer: $$\frac{1}{5}(x^6 - 3x^2)^{5} + C$$

Step-by-step solution

Identify the substitution function

We will choose a substitution function, u, which simplifies the given integral. In this case, let us choose u as the inner function: $$u = x^6 - 3x^2$$ Step 2: Calculate derivative of the substitution function

Find the derivative of u with respect to x

Now, we differentiate u with respect to x: $$\frac{du}{dx} = 6x^5 - 6x$$ Step 3: Express dx in terms of du

Find dx in terms of du

Rearrange the expression for the derivative to find dx: $$dx = \frac{du}{6x^5 - 6x}$$ Step 4: Substitute u and dx in the integral

Rewrite the integral in terms of u and du

Now we can rewrite the integral in terms of u: $$\int(x^{6}-3 x^{2})^{4}(x^{5}-x) d x = \int u^{4} (6x^5-6x) d (\frac{du}{6x^5 - 6x})$$ Step 5: Simplify the integral

Simplify the integral expression

The integral simplifies to: $$\int u^{4} du$$ Step 6: Calculate the integral

Evaluate the integral

We can now easily integrate the simplified expression: $$\int u^{4} du = \frac{1}{5}u^{5} + C$$, where C is a constant of integration. Step 7: Substitute back the original variable

Replace u with the original function

Now substitute the original function for u: $$\frac{1}{5}(x^6 - 3x^2)^{5} + C$$ Step 8: Check the result by differentiating

Differentiate the integrated function

To check our work, we differentiate the result: $$\frac{d}{dx} \left(\frac{1}{5}(x^6 - 3x^2)^{5} + C\right) = (x^{6}-3 x^{2})^{4}(x^{5}-x)$$

Final Answer

$$\int\left(x^{6}-3 x^{2}\right)^{4}\left(x^{5}-x\right) d x = \frac{1}{5}(x^6 - 3x^2)^{5} + C$$

Key concepts

Change of Variables

The concept of "change of variables" is a powerful technique used to simplify integrals by replacing a complex expression with a simpler one. This is achieved by introducing a new variable 'u' to take the place of some challenging parts of the integral. By doing so, the integral becomes more manageable and easier to solve.

To apply this technique, follow these steps:

• Identify the part of the integral that's complex or complicated. This is often inside a function or exponent.
• Define a new variable 'u' to represent this part. For example, in the given problem, we chose \(u = x^6 - 3x^2\) to simplify the expression.
• Express the differential \(du\) in terms of \(dx\), which involves differentiating \(u\) with respect to \(x\).

By substituting \(u\) and \(du\) back into the integral, you can convert a difficult integral into a simpler form that's easier to evaluate.

Substitution Method

The substitution method is a technique that leverages the change of variables concept to solve integrals. It's especially useful when direct integration is not clear or too complicated.

Here's how you proceed with substitution:

• After defining \(u\) and determining \(du\), you substitute \(u\) for its expression in terms of \(x\) in the integral.
• Your initial integral will transform into a straightforward expression involving \(u\) and \(du\).
• This simplification allows you to integrate the expression easily.

An important reminder is that once the integration is completed in terms of \(u\), you must substitute back \(x\) into your final expression to revert to the original variables. Applying this method converts the seemingly complex problem into simple algebraic operations.

Differentiation

Differentiation is the process of finding the derivative of a function, which tells us how a function changes as its input changes. In the context of checking indefinite integrals, differentiation is used to verify our integration results.

To differentiate effectively, understand these key points:

• After solving an integral, differentiate the result to ensure it matches the original function under the integral sign.
• Set the differential of the integrated expression in terms of \(x\) equal to the function originally provided in the integral.
• If these match, your integration is correct.

For the given problem, after integrating, we differentiate \(\frac{1}{5}(x^6 - 3x^2)^5 + C\) to check the result. Differentiation back to \((x^{6}-3 x^{2})^{4}(x^{5}-x)\) confirms that our integration process was successful.