Problem 39

Question

In Problems $37-42, a, b$, and $c$ are constants and $g(x)$ is a continuous function whose derivative $g^{\prime}(x)$ is also continuous. Use substitution to evaluate each indefinite integral. $$ \int g^{\prime}(x)[g(x)]^{n} d x $$

Step-by-Step Solution

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Answer

The integral is $ \frac{[g(x)]^{n+1}}{n+1} + C $.

1Step 1: Identify the Parts for Substitution

In the integral $ \int g'(x)[g(x)]^n \, dx $, we recognize that the function $ u $ can be $ u = g(x) $. This choice is useful because $ g'(x) \, dx $ becomes $ du $.

2Step 2: Perform the Substitution

Substitute $ u = g(x) $ and $ du = g'(x) \, dx $. The integral $ \int g'(x)[g(x)]^n \, dx $ now turns into $ \int u^n \, du $.

3Step 3: Integrate with Respect to u

The integral $ \int u^n \, du $ is a basic power rule integration problem. Using the power rule, $ \int u^n \, du = \frac{u^{n+1}}{n+1} + C $, where $ C $ is the constant of integration.

4Step 4: Substitute Back to x

Replace $ u $ with the original function $ g(x) $ to get the final answer in terms of $ x $. Thus, $ \frac{u^{n+1}}{n+1} + C $ becomes $ \frac{[g(x)]^{n+1}}{n+1} + C $.

Key Concepts

Indefinite IntegralPower Rule for IntegrationContinuous Functions

Indefinite Integral

An indefinite integral is a fundamental concept in calculus that is essentially the reverse process of differentiation. Think of it as an attempt to find a function whose derivative is the given integrand. In simple terms, it is about retrieving the original function from its derivative.

The symbol for indefinite integrals is $ \int $. That elongated 'S' shape indicates the process of integration.
The goal is to determine what function the integrand could have been derived from. Therefore, it does not have specific limits or bounds, unlike definite integrals which do.
Every indefinite integral comes with a constant of integration, denoted as $ C $, because there are infinitely many functions that could differ by a constant and still have the same derivative.

For example, for the integral $ \int 2x \, dx $, you'll find that it equals $ x^2 + C $. Here, $ C $ is crucial because without it, you couldn't account for all possible solutions. Indefinite integrals are thus used frequently to determine the original function from a rate of change.

Power Rule for Integration

The power rule for integration is one of the most straightforward techniques for finding indefinite integrals, especially when dealing with polynomials. This rule allows you to integrate functions that look like powers of $ x $.

The basic formula is: $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $, where $ n eq -1 $.
Essentially, you increase the power of $ x $ by one and then divide by the new power.
The constant $ C $ is again included to account for any additional constants that might have been present before differentiation.

Let’s see it in action: the integral $ \int x^3 \, dx $ would be computed as $ \frac{x^{4}}{4} + C $. It’s important to remember the condition $ n eq -1 $, because that requires a different integration technique (related to logarithmic integration). This rule is often your go-to solution for polynomial expressions, making it immensely useful in many calculus problems.

Continuous Functions

Understanding the nature of continuous functions is key to applying integration effectively. A function is said to be continuous if, informally speaking, you can draw it without lifting your pencil from the paper.

Continuous functions do not have gaps, jumps, or holes in their graphs.
This property ensures that the function behaves predictably over its domain: its output only changes gradually.
Both the function and its derivative need to be continuous to ensure smooth integration processes.

In the exercise we discussed, $ g(x) $ was a continuous function with a continuous derivative, $ g'(x) $. This allows us to apply integration techniques without worrying about irregularities in the function’s behavior. Continuous functions reveal crucial information about accumulation and areas under curves, making them indispensable in both theoretical and applied calculus.

Problem 39

Problem 40

Other exercises in this chapter

Problem 39

In Problems $39-48$, first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int e^{\sqrt{x}} d x $$

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Problem 39

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{3 x^{2}-x+1}{x(x-1)^{2}} d x $$

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Problem 40

(a) Show that $$0 \leq \frac{1}{\sqrt{x+x^{4}}} \leq \frac{1}{x^{2}}$$ for $x>0$. (b) Use your result in (a) to show that $$\int_{1}^{\infty} \frac{1}{\sqrt{x

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Problem 40

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{4 x^{2}+3 x+1}{(x+1)^{2}(x-1)} d x $$

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