Problem 47 - Calculus: Early Transcendental Functions

Question

Make the indicated substitution for an unspecified function \(f(x)\). $$u=\sin x \text { for } \int_{0}^{\pi / 2}(\cos x) f(\sin x) d x$$

Step-by-Step Solution

Answer

The simplified integral after substitution is \(\int_{0}^{1} f(u) du\).

Step 1: Identify the indicated substitution

Let \(u = \sin x\). Consequently, \(du = \cos x \ dx\).

Step 2: Change the limits of the integral

Since the substitution \(u = \sin x\) is made, the limits of the integral will change accordingly. When \(x = 0\), \(u = \sin 0 = 0\), and when \(x = \frac{\pi}{2}\), \(u = \sin \frac{\pi}{2} = 1\). Thus, the integral now moves from 0 to 1.

Step 3: Apply the substitution in the integral and simplify

The given integral \( \int_{0}^{\pi / 2}(\cos x) f(\sin x) d x\) becomes \(\int_{0}^{1} f(u) du\) after making the substitution \(u = \sin x\) and \(du = \cos x \ dx\)

Key Concepts

Integral Limits Change

When performing integration by substitution, altering the integral limits is a crucial step that follows naturally from the substitution itself. In the example provided, the substitution was from \( x \) to \( u=\sin x \). For such a trigonometric expression, it's important to adjust the limits of integration to reflect this transformation.

Original definite integrals, specified from \( x_1 \) to \( x_2 \), need the limits converted to new limits in terms of \( u \).

• When \( x = 0 \), substituting into \( \sin x \) gives \( u = \sin 0 = 0 \).
• When \( x = \frac{\pi}{2} \), substituting into \( \sin x \) gives \( u = \sin \frac{\pi}{2} = 1 \).

Using the substitution, the integral now ranges from \( u = 0 \) to \( u = 1 \). This conversion of limits is essential for evaluating the new integral correctly after substitution.

Trigonometric Substitution

Trigonometric substitution is a powerful technique used in calculus to simplify integrals involving tricky trigonometric parts. The example involves substituting \( u = \sin x \), making it much easier to evaluate the integral. Let's look at why and how this works.

Why Use Trigonometric Substitution?

• Trigonometric identities are often more straightforward to manipulate.
• They can simplify derivatives and integrals into forms we know how to tackle.

Steps

• Choose an appropriate substitution that simplifies the integral.
• Differentiate the substitution to relate \( du \) to \( dx \). For example, \( du = \cos x \ dx \).
• Change limits of the integral if it's a definite integral (as we did from 0 to 1 above).
• Substitute and Integrate: Replace all occurrences in the integral with the new variable \( u \), and integrate.

Such substitution transforms the problem into a simpler form, reducing the complexity of the integration process.

Indefinite Integrals

Indefinite integrals represent a broad class of functions as antiderivatives. Unlike definite integrals, they don't have limits of integration and instead, result in a family of functions plus a constant \( C \). Understanding them fundamentally aids in tackling both definite and indefinite integral problems.

Why Indefinite Integrals?

• They allow discovering families of functions where each has the same derivative.
• They lay the foundation for evaluating definite integrals through the Fundamental Theorem of Calculus.

Important Aspects of Indefinite Integrals

• General Form: The indefinite integral of \( f(x) \) is written as \( \int f(x) \ dx = F(x) + C \).
• Antiderivative: An antiderivative \( F(x) \) is a function whose derivative is \( f(x) \). The "+ \( C \)" denotes that any constant can be added, and it would still differentiate back to \( f(x) \).

Indefinite integrals play a key role in calculus, particularly when combined with substitution methods, to simplify and solve complex integrals into known, easier forms.