Problem 14 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-20\) , find the indefinite integral. $$ \int \frac{\sin x}{7+\cos ^{2} x} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the given function is \(-\ln |7+\cos^2 x| + C\).

Step 1: Identify the Form

This integral is in the form of \(\int \frac{f'(x)}{f(x)} dx\). In this case, \(f(x)\) is \(7+\cos ^{2} x\), whose derivative, \(f'(x)\), is \(-2\cos x \sin x\). Now, we can't immediately use the \(\sin x\) in the numerator as the derivative, as the \(-2\cos x\) factor is missing. We can, however, manipulate the integral to adjust for this.

Step 2: Manipulate the integral

Multiplying and dividing the integral by -2, we get \(-2 \int \frac{-\sin x}{7+\cos ^{2} x} dx\). Now, this looks similar to what we discussed in Step 1, but we need \(-\sin x \cos x\) in the numerator. Luckily, we can use the double-angle formula \(\sin 2x = 2\sin x \cos x\). Hence we can write \(-\sin x = \frac{\sin 2x}{2\cos x}\). So, our integral becomes \(-2 \int \frac{\sin 2x}{2\cos x(7 + \cos^2 x)} dx\). The 2 in the numerator and denominator will cancel out.

Step 3: Apply substitution

We can now substitute \(u = 7+\cos^2 x\). Hence, \(du = -2\cos x \sin x dx = -\sin 2x dx\). Therefore, our integral now becomes \(- \int \frac{du}{u}\).

Step 4: Compute the integral

The integral \(\int \frac{du}{u}\) is the natural log, so our integral becomes \(-\ln |u| + C\).

Step 5: Substitute back the original variable

Substitute \(u\) back into the integral to get the final answer: \(-\ln |7+\cos^2 x| + C\).

Key Concepts

Integration Techniques

Integrating functions often requires us to employ a variety of techniques to simplify and solve them. Among these methods, recognizing the form of the integral is crucial. In the given problem, the goal is to find the indefinite integral \( \int \frac{\sin x}{7+\cos ^{2} x} \, dx \).

This integral initially doesn't fit any basic formula, so we need a strategy to transform it. Techniques such as substitution, partial fraction decomposition, or recognizing trigonometric identities can help us rewrite the integral in a more solvable form.

Here, the focus is on manipulating the integrand into a form that allows substitution by using known trigonometric identities and algebraic manipulation. This approach aids in achieving an integral that is straightforward to evaluate.

Substitution Method

The substitution method is a powerful tool in integration. It involves changing the variable of integration to simplify the problem.

In this example, recognizing that the expression \(7+\cos^2 x\) inside the integral could be a candidate for substitution is the key. By letting \(u = 7+\cos^2 x\), the derivative \(du = -2\cos x \sin x \, dx = -\sin 2x \, dx\) emerges naturally, fitting well with the integral's structure.

Substitution transforms the given integral into a simpler, more easily integrable form: \(- \int \frac{du}{u}\). This substitution simplifies the integration process significantly, turning a complex-looking problem into a basic natural logarithm function.

Trigonometric Integration

Integrals involving trigonometric functions often require special techniques. Here, the integral deals with \(\sin x\) and \(\cos^2 x\).

The strategy employed involves using a double-angle identity: \(\sin 2x = 2\sin x \cos x\). This identity helps in expressing the integrand in terms of \(\sin 2x\), providing a pathway to match the substitution requirements.

This manipulation allows us to express \(-\sin x\) in terms of \(\sin 2x\), adjusting the integral so it aligns conveniently with our substitution approach. Using trigonometric identities is often necessary in integrals involving sines and cosines to transform the integrand into a more workable form.

Calculus

Calculus ties together various mathematical techniques and concepts, providing tools to solve complex integrals such as this one. Understanding both the theory and application of integration allows us to transition from seemingly unsolvable integrals to clear solutions.

By decomposing the integral \( \int \frac{\sin x}{7+\cos ^{2} x} \, dx \) into manageable parts through substitution and trigonometric integration, calculus demonstrates its strength.

The solution concludes with the integration yielding \(-\ln |u| + C\), where \(u = 7+\cos^2 x\). Substituting back gives the final answer \(-\ln |7+\cos^2 x| + C\). Calculus not only provides a structured approach to solve problems but also emphasizes the importance of creativity in applying mathematical identities and transformations.