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Chapter 8: Problem 9

Calculate. (If you run out of ideas, use the examples as models.) $$\int \sec ^{2} \pi x d x$$.

Short Answer

Expert verified
The short answer based on the given step-by-step solution is: \(\int \sec^{2}(\pi x) dx = \frac{1}{\pi}\tan(\pi x) + C\).

Step by step solution

01

Recognize the Derivative of the Tangent Function

First, we must recognize that the integrand here is the derivative of the tangent function with a slight modification: \(\frac{d}{dx}(\tan(\pi x)) = \pi\sec^2(\pi x)\).
02

Apply the Chain Rule

To make the integrand resemble more closely the derivative of the tangent function, we can introduce a constant factor of \(\pi\) inside the integral and another constant factor of \(\frac{1}{\pi}\) outside the integral: \(\frac{1}{\pi}\int \pi \sec^{2}(\pi x) dx\). With this modification, we can now apply the chain rule and see that: \(\frac{d}{dx}(\tan(\pi x)) = \pi\sec^2(\pi x)\).
03

Find the Antiderivative

Now that we have an integrand that is the derivative of a tangent function with respect to \(x\), we can find the antiderivative: \(\frac{1}{\pi} \int \pi \sec^{2}(\pi x) dx = \frac{1}{\pi} \tan(\pi x) + C\). Here, \(\tan(\pi x)\) is the antiderivative of \(\sec^2(\pi x)\) and \(C\) represents the constant of integration.
04

Write the Final Answer

We have now found the antiderivative of the given function: \(\int \sec^{2}(\pi x) dx = \frac{1}{\pi}\tan(\pi x) + C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integral Calculus
Integral calculus is a fundamental branch of calculus that focuses on integrals. While differential calculus involves derivatives, integral calculus deals with the integration, which is the inverse process of differentiation.
The main goal of integral calculus is to find the antiderivative or integral of a given function. This process allows us to determine the area under a curve, among many other applications.
In our exercise, we calculated the integral of \( \sec^2(\pi x) \), a function known to have a straightforward antiderivative.
Antiderivative
An antiderivative, or integral, of a function is another function whose derivative returns the original function.
For the function \( \sec^2(\pi x) \), its antiderivative is \( \tan(\pi x) \) because the derivative of \( \tan(\pi x) \) is \( \pi \sec^2(\pi x) \).
Remember to add a constant \( C \) to the antiderivative to account for any vertical shifts that won鈥檛 show up in the derivative, since differentiating a constant results in zero.
Chain Rule
The chain rule is a key technique in calculus used when differentiating compositions of functions. Conversely, it also plays a part in integration.
In the context of our exercise, applying the chain rule involved handling the \( \pi \) factor that appeared in the derivative of the tangent function.
To bring the integral into a form that permits straightforward integration, the expression \( \frac{1}{\pi}\int \pi \sec^2(\pi x) dx \) is used. This step simplifies integration similar to reversing the chain rule.
By doing so, the integration process became easier as it directly aligned with recognizing \( \tan(\pi x) \)'s derivative.
Integration Techniques
Integration techniques help solve complex integrals more efficiently. Some common techniques include substitution and integration by parts.
In our exercise, recognizing \( \sec^2(x) \) as a fundamental integral was vital. Such familiarity makes solving integrals direct and often independent of more elaborate methods.
However, recognizing when to tweak the coefficients, like multiplying and dividing by \( \pi \) to simplify the integral, is crucial and is where understanding techniques such as the chain rule plays a role.

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Most popular questions from this chapter

Use a graphing utility to draw the curve \(y=x \sin x\) for \(x \geq 0 .\) Then use a CAS to calculate the area between the curve and the \(x\) -axis (a) from \(x=0\) to \(x=\pi\) (b) from \(x=\pi\) to \(x=2 \pi\) (c) \(\operatorname{from} x=2 \pi\) to \(x=3 \pi\) (d) What is the area between the curve and the \(x\) -axis from \(x=n \pi\) to \(x=(n+1) \pi ?\) Take \(n\) an arbitrary nonncgative integer.

Let \(\Omega\) be the region under the curve \(y=\sqrt{x^{2}-a^{2}}\) from \(x=a\) to Find the volume of the solid generated by revolving \(\Omega\) about the \(x\) -axis and determine the centroid of that solid.

Evaluate. $$\int_{0}^{2} \frac{x}{x^{2}+5 x+6} d x$$

Calculate. (If you run out of ideas, use the examples as models.) $$\int_{0}^{2 / 4} \tan ^{3} x \sec ^{2} x d x$$.

Set $$f(x)=\frac{x}{x^{2}+5 x+6}$$,(a) Use a graphing utility to draw the graph of \(f\) (b) Calculate the area of the region that lies between the graph of \(f\) and the \(x\) -axis from \(x=0\) to \(x=4\).
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