Problem 63 - Calculus Early Transcendentals

Question

Use a change of variables to evaluate the following integrals. $$\int \sec ^{2} 10 x d x$$

Step-by-Step Solution

Answer

Question: Use a change of variables to evaluate the integral of \(\sec^2{(10x)}\) with respect to \(x\). Answer: \(\int \sec ^{2} 10 x d x = \frac{1}{10}\tan{(10x)} + C\)

Step 1: Identify the proper substitution

To make the integral easier to solve, we can use the substitution method. We will substitute \(u = 10x\). Now, differentiate \(u\) with respect to \(x\) to find \(d u / d x\). This will give us: $$\frac{d u}{d x} = 10$$ Next, solve for \(d x\) to use it in the integral: $$d x = \frac{d u}{10}$$

Step 2: Substitute the new variable in the integral

Now, replace \(10x\) with \(u\) and \(d x\) with \(d u/ 10\) in the original integral: $$\int \sec ^{2} 10 x d x = \int \sec^2{u} \cdot \frac{d u}{10}$$

Step 3: Evaluate the new integral

Evauating the integral, we obtain: $$\frac{1}{10}\int \sec^2{u} d u$$ Recall that the antiderivative of \(\sec^2{u}\) is \(\tan{u}\): $$\frac{1}{10}\int \sec^2{u} d u = \frac{1}{10}\tan{u} + C$$

Step 4: Substitute back the original variable

In the last step, we substitute back to the original variable \(x\). Substitute \(u = 10x\) in the expression obtained above: $$\frac{1}{10}\tan{u} + C = \frac{1}{10}\tan{(10x)} + C$$ So, the final answer is: $$\int \sec ^{2} 10 x d x = \frac{1}{10}\tan{(10x)} + C$$

Key Concepts

Substitution Method

The substitution method is a valuable technique in calculus for simplifying complex integrals. The core idea is to replace a part of the integral with a new variable, which makes it easier to evaluate. Here, it works like magic by transforming the integral into a friendlier form.
To use this method, identify a portion of the integral that, if 'changed,' could streamline the calculation process. Set this portion equal to a new variable, often denoted as \(u\). For instance, if we have an integral with variable \(x\), we might choose \(u = 10x\), like in our example.
Once the substitution is chosen, differentiate \(u\) with respect to \(x\) to find \(\frac{d u}{d x}\). This step is crucial because it allows us to express \(d x\) in terms of \(d u\). In our example, \(d x = \frac{d u}{10}\).
Afterward, substitute both the original expression and the variable \(d x\) with the new expressions involving \(u\). Now, the integral looks new and hopefully easier to tackle. At the end, don't forget to substitute back the original variable to get your final answer.

Trigonometric Integrals

Trigonometric integrals are integrals that involve trigonometric functions such as \(\sin\), \(\cos\), \(\tan\), \(\sec\), and others. They often appear intimidating, but once mastering a few techniques, they become much more manageable.
In many cases, knowledge of trigonometric identities and antiderivatives can be instrumental. For instance, the integral \(\int \sec^2 u \, du\) is one that ought to be memorized along with others like \(\int \tan u \, du\), and \(\int \sin u \, du\).
These basic antiderivatives form the foundation for solving more complicated trigonometric integrals. Recognize patterns in the integrals that match these known basic forms can simplify your work substantially. In this specific case, recognizing that the integral of \(\sec^2 u\) is \(\tan u\) allowed the problem to be solved swiftly once substitution was performed and \(u\) was in place of \(10x\).
Also navigate complexities within trigonometric identities to transform integrals into more familiar shapes, making them easier to solve.

Antiderivatives

The antiderivative is the reverse process of differentiation. If differentiating a function gives us another function, the antiderivative operation finds the original function that was differentiated.
At the core, if \(F(x)\) is an antiderivative of \(f(x)\), then \(F'(x) = f(x)\). So, computing antiderivatives is effectively asking, "What function, when differentiated, would give you \(f(x)\)?"
Antiderivatives often need to include a constant \(+ C\), because differentiating a constant yields zero, making it impossible to determine what, if any, constant was present in \(F(x)\).
In the context of our initial problem, we utilized the knowledge that the antiderivative of \(\sec^2 u\) is \(\tan u\). This is applied by first transforming our function into a simpler form through substitution and then integrating, or finding the antiderivative, within that simpler form.
When completing the integration and finding the antiderivative, always substitute back to your original variable (here, \(x\)) to finish solving the integral.