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Magazine Chapter 9: Problem 10
Determine the integrals by making appropriate substitutions. \(\int\left(x^{3}-6 x\right)^{7}\left(x^{2}-2\right) d x\)
Short Answer
Expert verified
\( \frac{(x^3 - 6x)^8}{24} + C \)
Step by step solution
01
Identify the Substitution
To simplify the integral, look for a substitution that will make the integrand easier to integrate. We notice that the derivative of the inner function \(u = x^{3} - 6x\) appears in the integrand. Specifically, set \(u = x^{3} - 6x\).
02
Calculate the Derivative of the Substitution
Find \frac{du}{dx}\: \[ \frac{du}{dx} = 3x^{2} - 6 \].
03
Express \dx\ in terms of \du\
Solve for \dx\: \[ du = (3x^2 - 6) \, dx \] \[ dx = \frac{du}{3x^2 - 6} \].
04
Simplify the Integral
Substitute \(u\) and \(dx\) into the integral: \[ \int (x^{3} - 6x)^{7}(x^{2} - 2) dx = \int u^{7}(x^{2} - 2) \frac{du}{3x^2 - 6} \].
05
Simplify Further
Notice that \(3x^2 - 6 = 3(x^2 - 2)\). Substitute that value in: \[ \int u^{7} \frac{x^{2} - 2}{3(x^2 - 2)} du = \int \frac{1}{3} u^7 du \].
06
Integrate
Now integrate \(\int \frac{1}{3} u^7 du\): \[ \frac{1}{3} \int u^7 du = \frac{1}{3} \cdot \frac{u^8}{8} + C = \frac{u^8}{24} + C \].
07
Back Substitution
Substitute back \(u = x^3 - 6x\): \[ \frac{(x^3 - 6x)^8}{24} + C \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
In integral calculus, the substitution method is a fundamental technique used to simplify integrals. This approach involves substituting a part of the integrand (the function being integrated) with a new variable, usually denoted as 'u'. When appropriately chosen, this substitution can transform a complex integral into a simpler one. The key is to identify a function within the integrand whose derivative is also present. For example, in the exercise, we chose the substitution \( u = x^3 - 6x \) because its derivative, \( 3x^2 - 6 \), is closely related to terms in the integrand.
Integrand
The term 'integrand' refers to the function that is being integrated in an integral. In the problem given, \( (x^3 - 6x)^7 (x^2 - 2) \) is the integrand. Simplifying this integrand is crucial for solving the integral. When you choose a substitution like \( u = x^3 - 6x \), you aim to rewrite the integrand in terms of the new variable, \( u \). This makes integration more manageable. Here, after substitution, the integral is expressed in terms of \( u \), simplifying the original complex function to a more straightforward form.
Integration by Substitution
Integration by substitution builds on the substitution method and involves both the substitution and integration processes. In our example:
This reformation often reduces a complicated integrand to a simpler form. For instance, \( (x^3 - 6x)^7 (x^2 - 2) dx \) becomes \( \frac{1}{3} u^7 du \), which is easier to integrate.
- First, we set \( u = x^3 - 6x \)
- Second, compute the derivative: \( \frac{du}{dx} = 3x^2 - 6 \)
- Third, solve for \( dx \): \( dx = \frac{du}{3x^2 - 6} \)
- Fourth, substitute \( u \) and \( dx \) back into the original integral:
This reformation often reduces a complicated integrand to a simpler form. For instance, \( (x^3 - 6x)^7 (x^2 - 2) dx \) becomes \( \frac{1}{3} u^7 du \), which is easier to integrate.
Integral Simplification
Integral simplification is an essential step in solving integrals efficiently. By carefully choosing an appropriate substitution, the integrand is simplified. In our case:
After substitution, the integral \( \frac{1}{3} \frac{u^7}{(x^2 - 2)} \frac{du}{3(x^2 - 2)} \) further simplifies to \( \frac{1}{3} u^7 du \). Performing this simplification makes the integration process straightforward. Finally, don't forget to substitute back in terms of the original variable to complete the solution, resulting in an easily understandable form: \( \frac{(x^3 - 6x)^8}{24} + C \).
- Recognize that \( 3x^2 - 6 \) can be factored as \( 3(x^2 - 2) \)
- This insight allows us to rewrite and simplify the integral effectively
After substitution, the integral \( \frac{1}{3} \frac{u^7}{(x^2 - 2)} \frac{du}{3(x^2 - 2)} \) further simplifies to \( \frac{1}{3} u^7 du \). Performing this simplification makes the integration process straightforward. Finally, don't forget to substitute back in terms of the original variable to complete the solution, resulting in an easily understandable form: \( \frac{(x^3 - 6x)^8}{24} + C \).
