Question

Find the indefinite integral and check the result by differentiation. $$ \int 2 x\left(x^{2}-1\right)^{7} d x $$

Short answer

The indefinite integral of \(2x(x^{2} - 1)^{7}\) with respect to \(x\) is \(\frac{1}{8}(x^{2} - 1)^{8} + C\).

Step-by-step solution

Identify the function and its derivative

In the given integral, it can be identified that the product of \(2x\) and the expression \((x^{2} - 1)^{7}\) resembles a function and its derivative setup. Peaking forward, it can be seen that \(2x\) is the derivative of \(x^{2}\). Therefore, it makes intuitive sense to let \(u = x^{2} - 1\). The differential of \(u\) with respect to \(x\) is \(du = 2x \, dx\).

Substitute function and its derivative

Substitute \(u = x^{2} - 1\) and \(du = 2x \, dx\). The integral changes to \(\int u^{7} \, du\), which is a simple power-rule integral.

Evaluate the integral

Evaluate the integral using the power rule for integrals, namely \(\int u^n \, du = \frac{1}{n+1}u^{n+1} + C\). Here, \(n = 7\), thus the result is \(\frac{1}{8}u^{8} + C\).

Substitute back

Now, substitute \(u = x^{2} - 1\) back into the equation. The integral then becomes \(\frac{1}{8}(x^{2} - 1)^{8} + C\)

Verify by differentiation

Verify the answer by differentiating \(\frac{1}{8}(x^{2} - 1)^{8} + C\). The derivative process will involve the application of the chain rule, since there is a composition of functions. The derivative results in \(2x(x^{2} - 1)^{7}\), which correctly matches the original integrand confirming that the integral setup and integration is correct.

Key concepts

Integration by Substitution

The integration by substitution method is a powerful tool for solving integrals. This technique works by simplifying an integral into a more manageable form. Here, you substitute part of the integrand with a new variable, usually denoted as \( u \). The goal is to transform the integral into one you can solve more easily.

In the exercise example, the expression \( 2x(x^2 - 1)^7 \) appears quite complex at first. We can see that \( 2x \) is the derivative of \( x^2 \), which suggests a substitution. We set \( u = x^2 - 1 \). Now, finding \( du \) involves differentiating \( u \) with respect to \( x \), leading to \( du = 2x \, dx \). With this step, the integral becomes \( \int u^7 \, du \), a much simpler problem.

Substitution is especially helpful because it converts potentially daunting expressions into their simpler forms, using basic rules of integration. Remember, when applying this technique, carefully identify a part of the integrand whose derivative is also present in the integral. This is key to a successful execution of the substitution method.

Power Rule for Integrals

The Power Rule for Integrals is one of the fundamental tools in calculating integrals. It states that for any function \( u^n \), where \( n eq -1 \), the integral is expressed as:

• \( \int u^n \, du = \frac{1}{n+1}u^{n+1} + C \)

This formula is straightforward and highly useful when working with polynomials, or when polynomials appear after substitution.

In the discussed problem, after substituting and simplifying, the integral transformed into \( \int u^7 \, du \). Using the Power Rule, we integrate it as \( \frac{1}{8}u^8 + C \). The constant \( C \) accounts for any constant value that may have been present in the original integrand but isn't directly visible during indefinite integration.

Always remember the Power Rule when encountering powers in integrals, as it significantly simplifies the process of finding solutions.

Verification by Differentiation

Verification by differentiation is crucial for confirming the correctness of your indefinite integral solution. After obtaining an integral solution, differentiating it should lead you back to the original integrand. This process assures that the integration was performed correctly and completely.

In the given exercise, after finding the indefinite integral as \( \frac{1}{8}(x^2 - 1)^8 + C \), we verify by differentiating. Applying the Chain Rule for differentiation, which handles composite functions, we derive the result:

• The outer function's derivative: \( 8 \times \frac{1}{8}(x^2 - 1)^7 = (x^2 - 1)^7 \)
• Multiply by the derivative of the inner function \( x^2 - 1 \), which is \( 2x \).

Combining these steps, we find that the derivative, \( 2x(x^2 - 1)^7 \), matches the original integrand. This confirms that our integration process was correctly executed and gives confidence in the accuracy of our answer.