Problem 36 Evaluate. (Be sure to check by d... [FREE SOLUTION]

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Calculus and its applications

Bittinger, Marvin L. Ellenbogen, David J. Surgent

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 4: Problem 36

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

Short Answer

Expert verified

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

Step by step solution

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.

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 $.

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 $.

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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91影视

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

Short Answer

Step by step solution

Identify the Integral Formula

Perform Substitution

Simplify and Integrate

Differentiate to Check

Key Concepts

Substitution Method

Antiderivative

Chain Rule

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