Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 36
use formal substitution (as illustrated in Examples 5 and 6) to find the indefinite integral. $$ \int 3 x^{2}\left(1-x^{3}\right)^{2} d x $$
Short Answer
Expert verified
The indefinite integral is \(x^3 - x^6 + \frac{x^9}{3} + C\)
Step by step solution
01
Identify the substitution
After closely examining the integral, one can see that the derivative of \(x^3\) is \(3x^2\), which is present in the integral. Therefore, it makes sense to let \(u = x^3\), making \(du = 3x^2 dx\).
02
Rewrite the integral
Substituting \(u = x^3\) into the integral, the integral becomes:\[\int (1-u)^2 du\]Now this integral is a standard form that can be solved easily.
03
Expand the integrand
The next step is to expand \((1-u)^2\). This gives the integral:\[\int 1 - 2u + u^2 du\]which is easier to solve
04
Evaluate the integral
Now we can perform the integration easily:\[\int 1 - 2u + u^2 du = u - u^2 + \frac{u^3}{3} + C\]where C represents the constant of integration.
05
Substitute back the original variable
Now we substitute the original variable back into the integral. This gives us:\[u - u^2 + \frac{u^3}{3} + C = x^3 - (x^3)^2 + \frac{(x^3)^3}{3} + C = x^3 - x^6 + \frac{x^9}{3} + C\]
06
Simplify the expression
Finally, the expression is simplified to:\[x^3 - x^6 + \frac{x^9}{3} + C\]
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
U-Substitution
The U-Substitution technique is essentially the reverse process of the chain rule from differentiation and is a powerful method for simplifying integrals that involve compositions of functions. The fundamental idea is to choose a new variable, typically denoted as \(u\), that will simplify the integral when it is substituted in place of the original variable.
To effectively use U-Substitution, follow these steps:
To effectively use U-Substitution, follow these steps:
- Identify a part of the integral that looks like the derivative of another part.
- Set \(u\) to be a function present in the integral whose derivative is also present.
- Rewrite the integral in terms of \(u\), including replacing \(dx\) with \(du\).
- Integrate with respect to \(u\).
- Finally, substitute the original variable back in to express the antiderivative in terms of the original variable.
Integration Techniques
Integration is a fundamental tool in calculus, often requiring a combination of techniques to solve more complex integrals. Some widely used methods include:
- U-Substitution, as detailed above.
- Integration by parts, which is the counterpart of the product rule for differentiation.
- Partial fractions decomposition, used to integrate rational functions.
- Trigonometric substitution, for integrals involving square roots of expressions involving squares.
Power Rule for Integration
The Power Rule for Integration is a basic integration rule that is instrumental in finding the antiderivative of powers of \(x\). The rule states that the integral of \(x^n\) with respect to \(x\), where \(n\) is any real number except -1, is given by:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]The Power Rule is visibly applied on each term after expanding the \(u\)-substituted integral: \(1\), \(u\), and \(u^2\). Here, you increment the exponent by 1 and then divide by the new exponent. Applying this rule in conjunction with U-Substitution simplifies our integral considerably, allowing us to integrate term by term.
Constant of Integration
When computing an indefinite integral, the result represents a family of functions that differ by a constant. To account for this family, it is essential to add a \(C\), known as the Constant of Integration, to the result. This constant represents all possible values of the antiderivative at a particular point and is crucial because indefinite integrals do not have specified bounds, unlike definite integrals. The presence of the constant acknowledges that there are infinitely many antiderivatives for a given function, and our textbook exercise concludes with adding \(C\) to the integrated expression to signify the general antiderivative.
