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Magazine Chapter 7: Problem 15
Find the integrals in problems. Check your answers by differentiation. $$ \int 12 x^{2} \cos \left(x^{3}\right) d x $$
Short Answer
Expert verified
The integral is \( 4 \sin(x^3) + C \).
Step by step solution
01
Identify the integration technique
To solve \( \int 12x^2 \cos(x^3) \, dx \), notice that the expression inside the cosine function, \( x^3 \), suggests a substitution method might work best.
02
Choose and perform substitution
Let \( u = x^3 \). Then, \( du = 3x^2 \, dx \), or \( x^2 \, dx = \frac{1}{3} \, du \). Substitute these into the integral, yielding \( \int 12x^2 \cos(x^3) \, dx = \int 12 \cos(u) \frac{1}{3} \, du \).
03
Simplify the integral
Simplify the integral expression to \( \int 4 \cos(u) \, du \).
04
Evaluate the integral
Integrate \( \int 4 \cos(u) \, du \) to get \( 4 \sin(u) + C \), where \( C \) is the constant of integration.
05
Reverse substitution
Since \( u = x^3 \), substitute back to obtain \( 4 \sin(x^3) + C \).
06
Differentiate to check the answer
Differentiate \( 4 \sin(x^3) + C \) with respect to \( x \). Using the chain rule, \( \frac{d}{dx}[4 \sin(x^3)] = 12x^2 \cos(x^3) \), which matches the original integrand.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful technique in calculus, especially useful when dealing with integrals involving composite functions. Imagine you're trying to simplify a complex problem by breaking it down into more manageable parts. In essence, you're looking to replace a piece of the integral with a new variable to make the integration simpler.
Here鈥檚 how it works:
Here鈥檚 how it works:
- Identify the part of the integral that, when substituted, will simplify your problem. Typically, this is the inner function of a composite expression.
- For the given exercise, we chose to let \( u = x^3 \). This choice simplifies the original integral since \( x^3 \) is inside the cosine function, a pattern suggesting easier integration post-substitution.
- Next, differentiate this substitute, \( u \), with respect to \( x \) to find \( du \). For our example, differentiating gives \( du = 3x^2 \, dx \).
- Express \( dx \) in terms of \( du \) to proceed with the substitution, allowing \( x^2 \, dx = \frac{1}{3} du \) to replace part of the integral鈥檚 \( dx \) term.
Differentiation
Differentiation is the reverse process of integration. It involves computing the derivative of a function, thereby finding the rate at which it changes. Understanding differentiation is key to verifying solutions obtained through integration.
In our context:
In our context:
- After finding the integral, differentiation helps confirm that our result is correct. In this exercise, after calculating the integral to derive \( 4 \sin(x^3) + C \), we take its derivative.
- Applying differentiation requires recognition of functions and deriving each piece orderly, as demonstrated in obtaining \( \frac{d}{dx}[4 \sin(x^3)] \). This step shows how integral and derivative are interconnected.
- By showing the differentiation returns the original integrand \( 12x^2 \cos(x^3) \), it assures the accuracy of the integral.
Chain Rule
The chain rule is a fundamental principle of differentiation used when dealing with composite functions. It specifies how to take the derivative of a function that has another function nested inside it, much like our integral problem.
How the chain rule works:
How the chain rule works:
- When you have a composite function, such as \( \sin(x^3) \), the chain rule says to differentiate the outer function and then multiply it by the derivative of the inner function.
- In mathematical terms, if \( y = g(f(x)) \), then \( \frac{dy}{dx} = g'(f(x)) \times f'(x) \).
- For our integral check: To differentiate \( 4 \sin(x^3) \), first, differentiate \( \sin(x^3) \) resulting in \( \cos(x^3) \) and then multiply by \( 3x^2 \), which is the derivative of \( x^3 \). This complete application yields \( 12x^2 \cos(x^3) \).
