Problem 45 - Calculus: Early Transcendental Functions

Question

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

Step-by-Step Solution

Answer

The result of the substitution of \(u=x^{2}\) for the integral \( \int_{0}^{2} x f\left(x^{2}\right) dx\) is \( 1/2 \int_{0}^{4} f(u) du \).

Step 1: Substituting Given Values

Substitute \( u = x^2 \). This allows the rewriting of the function within the integral as \( f(u) \) instead of \( f(x^{2}) \). Hence, the integral becomes \( \int x f(u) dx \).

Step 2: Differentiating with Respect to x

Differentiate \( u=x^{2} \) with respect to \( x \) to get \( du = 2x dx \). Solve for \( dx \) to replace it in the integral by \( dx = du/(2x) \). As a result, the integral becomes \( \int x f(u) du/(2x) \).

Step 3: Simplified Integral Expression

Simplify the integral expression by cancelling out \( x \) from the numerator and denominator, and bring the constant 1/2 out of the integral. This results in \( 1/2 \int f(u) du \).

Step 4: Change of integration limits

Substitute the original limits of integration which were in terms of \( x \) into \( u = x^2 \) to get the new limits in terms of \( u \). Thus, \( u \) at \( x = 0 \) is 0 and at \( x = 2 \) is 4, leading to a new integral: \( 1/2 \int_{0}^{4} f(u) du \).

Key Concepts

Definite Integrals

Definite integrals are a foundational concept in calculus that help us find the total accumulation of a quantity, like area under a curve, over a specific interval.
Typically, when you compute a definite integral, you take two steps: evaluate the integral and substitute the limits of integration. What's unique about definite integrals is that they have upper and lower limits, often denoted as \(a\) and \(b\), giving the integral its definite nature.
Common uses of definite integrals include finding volume, displacement, and other physical applications. They are a powerful tool for measuring how quantities change over an interval, making them pivotal in both theoretical and applied mathematics.

Variable Substitution

Variable substitution, often called "u-substitution," is a powerful technique in calculus used to simplify complex integrals.
In this method, you replace a part of the function with a new variable, frequently denoted as \(u\), to make integration easier. The purpose of substitution is to transform a difficult integral into a simpler form.

• Choose a substitution: Begin by choosing a substitution that can simplify part of the integrand.
• Differentiate the substitution: Differentiate \(u\) with respect to \(x\) to express \(du\) in terms of \(dx\).
• Adjust limits of integration: If dealing with definite integrals, convert the original limits to the new variable \(u\).

This technique leverages the chain rule of differentiation, making the integration process more manageable.

Calculus Steps

When tackling integral problems, following a structured set of steps is essential to reach a solution effectively.
For the substitution method outlined, here's a quick recap of the calculus steps:

• Identify a substitution that can reduce the complexity of the integrand.
• Rewrite the integral using the substitution, expressing all necessary parts in the new variable.
• Perform differentiation to relate \(dx\) to \(du\), simplifying the integral further if possible.
• Simplify the expression, and note how constants factor into the integral.
• In definite integrals, replace the original interval endpoints with their corresponding new values, ensuring the limits match the substitution.
• Evaluate the new integral, substituting back to the original variable if needed.

These steps bring clarity to the process, helping to systematically solve integration problems.