Problem 27 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the indefinite integrals. $$ \int \frac{1}{x} \csc ^{2}(\ln x) d x $$

Step-by-Step Solution

Answer

The integral is \( -\cot(\ln x) + C \).

Step 1: Identify the Inner Function for Substitution

In the given integral \( \int \frac{1}{x} \csc^{2}(\ln x) \, dx \), we notice that \( \ln x \) is a natural choice for substitution, as it is within the \( \csc^2 \) function. Let \( u = \ln x \).

Step 2: Calculate the Differential

Calculate the differential of \( u = \ln x \). The derivative of \( \ln x \) with respect to \( x \) is \( \frac{1}{x} \). Therefore, \( du = \frac{1}{x} \, dx \).

Step 3: Substitute in Terms of \( u \)

Rewrite the original integral in terms of \( u \). The substitution gives us \( du = \frac{1}{x} \, dx \), so: \[ \int \csc^2(u) \, du \].

Step 4: Integrate in Terms of \( u \)

The integral \( \int \csc^2(u) \, du \) is a standard integral with the result \( -\cot(u) + C \), where \( C \) is the constant of integration.

Step 5: Substitute Back in Terms of \( x \)

Substitute \( u = \ln x \) back into the result to get the function in terms of \( x \). Therefore, the integral becomes \( -\cot(\ln x) + C \).

Key Concepts

Indefinite Integrals

When we talk about indefinite integrals, we are referring to the reverse process of differentiation. Essentially, finding an indefinite integral is about determining a function which, when differentiated, yields the original integrand. The notation for an indefinite integral is \( \int f(x) \, dx \). Unlike definite integrals, which give us a numerical value representing the area under a curve, indefinite integrals produce a family of functions. This family of functions is accompanied by a constant of integration, usually represented by \( C \).

• Antiderivative: The primary goal of finding an indefinite integral is determining the antiderivative, which is a function whose derivative matches the given integrand.
• The Constant \( C \): Since the process involves reversing differentiation, we need to account for the constant of integration \( C \), as differentiation of a constant yields zero, leaving it indistinguishable when working back to the integral.

An example of an indefinite integral may look like \( \int x^2 \, dx = \frac{x^3}{3} + C \), where \( \frac{x^3}{3} \) is the antiderivative and \( C \) is the constant. In problems with trigonometric or logarithmic functions, as seen in the provided exercise, techniques like substitution are often used to simplify the process of finding the antiderivative.

Functions of Natural Logarithm

Functions involving natural logarithms, denoted as \( \ln x \), frequently appear in calculus, particularly during integration and differentiation. A natural logarithm is based on the constant \( e \), where \( e \approx 2.71828 \), a fundamental number that plays a crucial role in calculus and real-world applications.

• Natural logarithm properties: The derivative of \( \ln x \) is \( \frac{1}{x} \).
• Substitution in Integrals: When the integrand includes expressions like \( \ln x \), substitution can simplify the integration process. For instance, in our exercise, we use \( u = \ln x \), making differentiation straightforward because it leads to \( du = \frac{1}{x} \, dx \).

The function of the natural logarithm is crucial in calculus due to its unique rate of growth and the property of reversing exponentiation. You will often encounter this function during integration steps that involve substitution methods. Its presence transforms seemingly complex integrals into manageable forms, as seen in the exercise solution where \( \ln x \) helped rewrite the integral as a standard trigonometric form: \( \int \csc^2(u) \, du \).

Trigonometric Integration

Trigonometric integration is a key technique in calculus used to solve integrals involving trigonometric functions like sine, cosine, tangent, and their reciprocals such as cosecant, secant, and cotangent. These integrals often rely on recognizing fundamental identities and formulas for integration.

• Common Integrals: Some standard integrals to remember include \( \int \sin x \, dx = -\cos x + C \), \( \int \cos x \, dx = \sin x + C \), and quite relevant to the given exercise, \( \int \csc^2 x \, dx = -\cot x + C \).
• Use of Identities: Trigonometric identities, like \( \csc^2(x) = 1 + \cot^2(x) \), can be employed to rework integrals into solvable forms.

In the exercise, once you've performed the substitution \( u = \ln x \), the integral \( \int \csc^2(u) \, du \) comes into play. Recognizing this as a common and easily integrable form aids in efficiently finding the solution. These techniques are beneficial in making complex integrals tractable, highlighting the versatility and necessity of mastering trigonometric identities in calculus.