Question

Verify the following indefinite integrals by differentiation. These integrals are derived in later chapters. $$\int x^{2} \cos x^{3} d x=\frac{1}{3} \sin x^{3}+C$$

Short answer

Question: Verify that the given indefinite integral result is correct: $$\int x^{2} \cos x^{3} dx = \frac{1}{3} \sin x^{3} + C$$ Answer: The derivative of the given integral result is equal to the initial integrand, which confirms that the provided indefinite integral result is correct: $$x^{2}\cos(x^{3})$$

Step-by-step solution

Identify the integral and its result

We are given an integral problem: $$\int x^{2} \cos x^{3} dx$$ And its result as: $$ \frac{1}{3} \sin x^{3} + C$$

Find the derivative of the integral result

We will differentiate the given integral result using Chain Rule. Chain Rule states that if we have a function of the form $$h(g(x))$$, then $$\frac{d}{dx}(h(g(x)))=h'(g(x))\cdot g'(x)$$ Apply the Chain Rule to our integral result: $$\frac{d}{dx}\left( \frac{1}{3}\sin(x^{3}) + C \right) = \frac{1}{3}\cdot\frac{d}{dx}\left(\sin(x^{3})\right) + \frac{d}{dx}(C)$$ As the derivative of the constant C vanishes, we can write it as: $$\frac{1}{3}\cdot\frac{d}{dx}\left(\sin(x^{3})\right)$$

Differentiate using Chain Rule

Now, we will differentiate the sine expression inside the derivative: $$\frac{1}{3}\cdot\frac{d}{dx}\left(\sin(x^{3})\right) = \frac{1}{3}\cdot(\cos(x^{3}))\cdot\frac{d}{dx}(x^{3})$$ The derivative of the cubic function \((x^{3})\) with respect to x is \(3x^{2}\). So, we have: $$\frac{1}{3}\cdot(\cos(x^{3}))\cdot3x^{2}$$

Simplify the expression

As we can see, the expression simplifies nicely: $$\frac{1}{3}\cdot(\cos(x^{3}))\cdot3x^{2}=x^{2}\cos(x^{3})$$

Verify the results

Our derivative is now equal to our initial integrand: $$x^{2}\cos(x^{3})$$ Thus, we have confirmed that the indefinite integral result is correct because when differentiating it, our answer matches the given integrand: $$\int x^{2} \cos x^{3} dx=\frac{1}{3} \sin x^{3}+C$$

Key concepts

Chain Rule

Understanding the chain rule is fundamental to both differentiation and integration. When presented with a composite function—one inside another, like \( h(g(x)) \)—the chain rule instructs us to take the derivative of the outer function and multiply it by the derivative of the inner function. In mathematical terms, the formula is:\
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\( \frac{d}{dx}(h(g(x))) = h'(g(x)) \cdot g'(x) \)\
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For instance, if we have a function \( y = \sin(x^3) \) and we wish to find its derivative, we recognize \( \sin( ) \) as the outer function and \( x^3 \) as the inner function. By applying the chain rule, we differentiate \( \sin( ) \) to get \( \cos( ) \) and multiply it by the derivative of \( x^3 \) which is \( 3x^2 \). This yields \( 3x^2 \cos(x^3) \) as the derivative of our original function. Understanding this concept allows us to move forward with confirming integrals by differentiation, as employed in the textbook exercise.

Differentiation

Differentiation is a technique used to calculate the rate at which a given function is changing at any given point. It is the process of finding the derivative of a function, which is essentially the function's instant rate of change or slope. Differentiation is used to analyze all sorts of changes and is crucial to mastering calculus.\
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For a function \( y = f(x) \), the derivative is commonly denoted as \( f'(x) \) or \( \frac{dy}{dx} \) and is calculated following specific rules for various types of functions. For example, the power rule states that if \( f(x) = x^n \) then \( f'(x) = nx^{n-1} \). By breaking down a function into simpler parts and systematically applying the relevant differentiation rules, we can determine the derivative of even the most complex functions.

Integral Verification by Differentiation

When tackling integrals, especially indefinite ones, it's often useful to verify solutions using differentiation. The principle behind integral verification by differentiation is straightforward: since integration and differentiation are inverse operations, taking the derivative of an integral should yield the original function that was integrated (apart from the constant of integration).\
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To implement this verification, we start with the proposed solution to an integral, differentiate it, and see if the result matches the original function to be integrated. As shown in the exercise, the proposed integral of \( x^2 \cos(x^3) \) was \( \frac{1}{3} \sin(x^3) + C \). By differentiating this expression and simplifying the result, if we obtain the original function, it confirms the correctness of the integral.

Cosine Function Integration

Integration of the cosine function, or any trigonometric function, follows specific techniques tailored to the properties of these functions. For the cosine function, its integral is the sine function, with necessary adjustments for any inner functions present according to the chain rule.\
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The integral of a cosine function such as \( \cos(x) \) is simply \( \sin(x) + C \) where \( C \) is the constant of integration. However, when dealing with a composite function like \( \cos(x^3) \) as seen in the textbook exercise, applying the chain rule is essential for successful integration. An additional factor, often requiring the reverse application of differentiation rules, may appear, such as a multiplicative constant that must be taken into account to obtain the correct antiderivative.