Problem 43 - Biocalculus Calculus for the Life Sciences

Question

Find the general indefinite integral. \(\int \frac{\sin x}{1-\sin ^{2} x} d x\)

Step-by-Step Solution

Answer

The indefinite integral is \(\sec x + C\).

Step 1: Recognize Trigonometric Identity

Firstly, identify that the denominator can be rewritten using the Pythagorean identity. We know that \(1 - \sin^2 x = \cos^2 x\). Thus, the integral becomes \(\int \frac{\sin x}{\cos^2 x} \, dx\).

Step 2: Simplify the Integral

Notice that \(\frac{\sin x}{\cos^2 x}\) can be rewritten as \(\sin x \cdot \sec^2 x\). We can express this as \(\int \sin x \cdot \sec^2 x \, dx\).

Step 3: Apply Trigonometric Substitution

Use the substitution method by letting \(u = \cos x\). Then, \(\frac{du}{dx} = -\sin x\) or \(du = -\sin x \, dx\). Substitute in the integral to get \(-\int \frac{1}{u^2} \, du\).

Step 4: Integrate the Simplified Expression

The integral \(-\int \frac{1}{u^2} \, du\) can be rewritten as \(-\int u^{-2} \, du\). Use the power rule of integration: \(\int u^n \, du = \frac{u^{n+1}}{n+1} + C\) for \(n eq -1\). Therefore, the integral becomes \(-\left(-\frac{1}{u}\right) = \frac{1}{u} + C\).

Step 5: Substitute Back the Original Variable

Replace \(u\) with the original expression in terms of \(x\). Therefore, since \(u = \cos x\), the integral becomes \(\frac{1}{\cos x} + C\), which is \(\sec x + C\).

Step 6: Conclusion: Write the General Indefinite Integral

The general indefinite integral of the function is \(\sec x + C\).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify certain integrals involving trigonometric functions. By using a trigonometric identity or substitution, complicated expressions can be transformed into simpler ones. In our example, we use the substitution method by letting a trigonometric function, such as cosine, replace another variable.

• This is helpful when you encounter expressions that contain terms like \(1 - ext{trig function}^2\), which can be linked to well-known trigonometric identities.
• For instance, when we see the expression \(1 - \sin^2 x\), it’s beneficial to substitute using identities that relate sine and cosine.
• Substitution simplifies the integral, making it easier to evaluate.

In our integral, we selected \(u = \cos x\) which allowed us to transform the original expression into one involving \(u\). This substitution simplifies calculations and helps achieve the solution.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry. It states that for any angle \(x\), the expression \(\sin^2 x + \cos^2 x = 1\).
This identity is particularly useful when simplifying integrals that have trigonometric functions.

• With it, you can replace expressions like \(1 - \sin^2 x\) to \(\cos^2 x\). This simplifies integrals significantly.
• Using this identity often turns a seemingly complex integral into a form that is much easier to work with.

In our case, \(1 - \sin^2 x\) was transformed to \(\cos^2 x\) using this identity, which made further transformation manageable.

Power Rule of Integration

The power rule of integration is an essential tool for integrating functions that are simple powers of variables. When you have the integral of the form \(\int u^n \, du\), you can use the power rule: \[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C, \quad \text{for } n eq -1 \]

• This rule helps to quickly find the antiderivative of polynomial expressions.
• It is straightforward, and its application is one of the foundational skills in integral calculus.

In our exercise, after using substitution, we reached the integral form \(-\int u^{-2} \, du\), which was simplified using the power rule to find the antiderivative. This simplification was key in moving towards the final solution.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are invaluable when transforming and simplifying integrals.

• Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities.
• These identities let us express functions in terms of each other to simplify complex problems. They're especially handy in integrals involving products or quotients of sine, cosine, etc.

In the given problem, trigonometric identities like \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec x = \frac{1}{\cos x}\) were used to express the integral in a manageable form. Our conclusion, \(\sec x + C\), shows the power of these identities in simplifying intricate trigonometric expressions.