Problem 36 - Calculus and its applications

Question

Evaluate. (Be sure to check by differentiating!) $$ \int \sqrt{a x+b} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{2}{3a} (ax+b)^{3/2} + C \).

Step 1: Identify the Integral Formula

We have the integral \( \int \sqrt{a x + b} \, dx \). This is similar to the form \( \int \sqrt{u} \, du \), which integrates to \( \frac{2}{3} u^{3/2} + C \). To proceed, we'll perform a substitution to simplify the expression.

Step 2: Perform Substitution

Let \( u = ax + b \). Then, \( du = a \, dx \) or \( dx = \frac{1}{a} \ du \). This substitution transforms the integral into \( \int \sqrt{u} \cdot \frac{1}{a} \, du \).

Step 3: Simplify and Integrate

The integral becomes \( \frac{1}{a} \int u^{1/2} \, du \). The antiderivative of \( u^{1/2} \) is \( \frac{2}{3} u^{3/2} \), so the integral evaluates to \( \frac{1}{a} \cdot \frac{2}{3} u^{3/2} + C = \frac{2}{3a} (ax + b)^{3/2} + C \).

Step 4: Differentiate to Check

Differentiate \( \frac{2}{3a} (ax + b)^{3/2} + C \) to ensure it gives the original integrand. Using the chain rule, the derivative is \( \frac{2}{3a} \cdot \frac{3}{2} (ax+b)^{1/2} \cdot a = \sqrt{ax + b} \), matching the original integrand.

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus that simplifies the process of evaluating complex integrals. It involves substituting a part of the integrand with a new variable, usually denoted as \( u \). This is particularly useful when the integral contains a composite function that can be expressed more easily with a simpler function.

• To apply the substitution method, identify a function and its derivative within the integral.
• Convert the integrand by substituting \( u = g(x) \), where \( g(x) \) is a part of the original function.
• Determine \( du \), which is the derivative of \( g(x) \). This step ensures that all elements of the integrand are in terms of \( u \) and \( du \).

In our exercise, the substitution \( u = ax + b \) leads us to rewrite the integral in terms of \( u \), simplifying the integration process. This method effectively reduces complex expressions to more manageable integrals.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the given function. Finding an antiderivative is the reverse process of differentiation. When we integrate a function, we are essentially looking for its antiderivative.

• The notation \( \int f(x) \, dx \) represents the antiderivative of \( f(x) \).
• Integration constants, typically denoted \( C \), play a crucial role, representing an infinite number of potential solutions.
• The power rule for antiderivatives is simple: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for any real number \( n eq -1 \).

In the original exercise, the integral \( \int u^{1/2} \ du \) represents a scenario where the resulting antiderivative is \( \frac{2}{3}u^{3/2} + C \). This illustrates how to find a function that, when differentiated, yields the initial integrand.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate compositions of functions. It is essential when verifying our integration work through differentiation, ensuring the accuracy of our antiderivative.

• The chain rule states: If a function \( y = f(g(x)) \) is composed of two functions \( f \) and \( g \), its derivative is \( f'(g(x)) \cdot g'(x) \).
• It allows for the differentiation of complex, nested functions that can't be straightforwardly separated.
• Applied in reverse, it helps in identifying the original integrand after finding an antiderivative.

In the solution provided, after obtaining the antiderivative \( \frac{2}{3a}(ax + b)^{3/2} + C \), the chain rule confirms the result. Differentiating with respect to \( x \) gives \( \sqrt{ax + b} \), validating the correctness of the solution by showing that our derived function's differentiation returns the original integrand. This ensures our integration process hasn't gone astray.