Problem 63 - Calculus: Early Transcendental Functions

Question

Find an antiderivative by reversing the chain rule, product rule or quotient rule. $$\int 2 x \cos x^{2} d x$$

Step-by-Step Solution

Answer

The antiderivative of \( 2x \cos(x^2) \) is \( \sin(x^2)/2 + C \).

Step 1: Identify the inner function for substitution

In the integral \( \int 2x \cos(x^2) dx \), we are going to identify the inner function that will be convenient for substitution. Here, the natural choice would be \( u = x^2 \).

Step 2: Compute the derivative of u

Next, the derivative of \( u = x^2 \) is computed as \( du = 2x dx \). So, \( dx = du /(2x) \).

Step 3: Substitute u and dx in the integral

Substitute \( u = x^2 \) and \( dx = du /(2x) \) in the integral, so that it becomes \( \int \cos(u) du /2 \).

Step 4: Compute the integral

Now, the integral simplifies to \( 1/2 \int \cos(u) du \). The integral of \( \cos(u) \) is \( \sin(u) \). Thus the simplified integral becomes \( 1/2 \sin(u) + C \) where C is the constant of integration.

Step 5: Substitute back for x

Finally, substitute back \( u = x^2 \) to get the final answer. Thus, the antiderivative of \( 2x \cos(x^2) \) is \( \sin(x^2)/2 + C \)

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate compositions of functions. It's especially useful when dealing with functions nested within each other.
For instance, if you have a function composed of another function like \( f(g(x)) \), the chain rule helps you find its derivative. The derivative of \( f(g(x)) \) is given by \( f'(g(x)) \cdot g'(x) \).

This rule is important because it allows us to break down complex derivatives into simpler parts. If you understand how to apply it, solving problems becomes much easier.

• Basic Idea: Differentiate the outer function and multiply by the derivative of the inner function.
• Example: If you have \( (3x^2 + 2)^5 \), the derivative is computed using the chain rule.
• Apply: Let \( u = 3x^2 + 2 \), then derivative is \( 5u^4 \cdot 6x \).

Applying the chain rule properly helps in understanding antiderivatives, because when reversing the differentiation process—known as integration—we often look to reassemble these simple derivatives into their original complex forms.

Integration Techniques

Integration techniques are strategies to solve integrals, which are the opposite process of differentiation. In calculus, integration helps in finding antiderivatives and calculating total values like area.

• Basic Integration: Directly integrate simple functions, such as \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \).
• Advanced Methods: Methods such as substitution and integration by parts are used for more complex integrals.
• Key Concept: Antiderivatives are found by reversing differentiation rules like chain, product, or quotient rules.

Understanding these techniques is key to solving complex calculus problems. They guide you in transforming an integral into a form that's easy to compute. It's like knowing different pathways to complete a puzzle; the more techniques you know, the faster you get to the solution. For example, substitution is a method that simplifies the form of an integral transforming it into a basic one. So, mastering these techniques is crucial for integration success.

Substitution Method

The substitution method is a powerful technique in integral calculus that simplifies integration.
This method works similar to the chain rule but in reverse, helping to simplify complex integrals by focusing on an inner function.

In essence, you're finding a way to transform the variable of integration into something more manageable.

• Start by identifying the inner function: Look for a part of the integral that if substituted will make it easier.
• Compute its derivative: Knowing this helps substitute the differential \( dx \) accordingly.
• Rewrite the integral: Transform the original variable into \( u \), the integral becomes much simpler.
• Integrate: With the new expression, perform the integration.
• Back-substitute: Finally, replace \( u \) with the original variable to find the solution.

Consider the original integral \( \int 2x \cos(x^2) \, dx \). Here the substitution \( u = x^2 \) was used, transforming the integral into \( \int \cos(u) \, du \).
This process makes it simpler to solve. Once integrated, replacing \( u \) back with \( x^2 \) yields the final solution.