Problem 6 Exer. $1-8$ : Evaluate the int... [FREE SOLUTION]

Chapter 5: Problem 6

Exer. $1-8$ : Evaluate the integral using the given substitution, and express the answer in terms of $x$. $$ \int \frac{1}{(5 x-4)^{10}} d x ; \quad u=5 x-4 $$

Short Answer

Expert verified

The integral is $ -\frac{1}{9(5x - 4)^9} + C $.

Step by step solution

Identify the Substitution

We are given the substitution $ u = 5x - 4 $. This substitution simplifies the expression inside the integral.

Find du/dx and dx

Differentiate $ u = 5x - 4 $ with respect to $ x $ to find $ \frac{du}{dx} $. Thus, $ \frac{du}{dx} = 5 $, which implies $ dx = \frac{du}{5} $.

Substitute Into the Integral

With $ u = 5x - 4 $ and $ dx = \frac{du}{5} $, replace $ (5x - 4) $ with $ u $ and $ dx $ with $ \frac{du}{5} $: \[ \int \frac{1}{u^{10}} \frac{du}{5} = \frac{1}{5} \int u^{-10} \ du \]

Integrate with Respect to u

Integrate $ u^{-10} $ term-by-term:\[ \int u^{-10} \ du = \frac{u^{-9}}{-9} + C = -\frac{1}{9u^9} + C \]

Substitute Back in Terms of x

Replace $ u $ back with $ 5x - 4 $ to express the integral in terms of $ x $:\[ -\frac{1}{9(5x - 4)^9} + C \]

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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 tool in integral calculus used to simplify the process of finding integrals. The idea is to transform a complicated integral into a simpler one that is easier to solve. This method involves substituting a part of the integral with a new variable, often making the integration process more straightforward.

Here's how it works:

Identify a part of the integral that can be replaced with a single variable. In our case, we have chosen to set $ u = 5x - 4 $.
Differentiate the substitution equation to express $ dx $ in terms of $ du $. For instance, if $ u = 5x - 4 $, then $ \frac{du}{dx} = 5 $ implies $ dx = \frac{du}{5} $.
Replace all occurrences of $ x $ in the original integral with $ u $ and $ dx $ with $ \frac{du}{5} $, which simplifies the problem significantly.

Definite Integral

A definite integral calculates the area under a curve within specified bounds. Unlike indefinite integrals, which represent a family of functions, definite integrals result in a single number.

When evaluating definite integrals using the substitution method, it鈥檚 crucial to update the limits of integration to reflect changes in the variable. Here's a quick rundown:

Apply the substitution to transform both the integrand and the limits of integration.
After integration, if you substituted a variable, make sure to change your answer back to the original variable. This ensures that the final result truly reflects the original problem setup.
The final result of a definite integral can also be interpreted as the net area bounded by the curve within the limits.

Indefinite Integral

Indefinite integrals represent the most general form of antiderivatives. They denote a family of functions plus a constant of integration $ C $. This constant arises because integrating a function reverses the differentiation process, and differentiation of a constant is zero.

Key points to understand about indefinite integrals:

An indefinite integral is essentially asking, "What function, when differentiated, results in the original function inside the integral?"
The constant of integration $ C $ captures any constant term that would disappear through differentiation.
In our solved example, integrating $ u^{-10} $ results in an expression that includes this constant $ C $, indicating there are many possible functions that could fit this integral.

Derivative Differentiation

Differentiation is the mathematical process of finding a derivative, which measures how a function changes as its input changes. It's a fundamental concept in calculus, providing insight into rates of change and slopes of curves.

Differentiation is closely linked to integration, as they are essentially inverse operations:

In the substitution method, we use differentiation to transform expressions 鈥� for example, finding $ \frac{du}{dx} $ helps in substituting the variable for integration.
Understanding how to differentiate correctly is crucial for accurate integration, as it sets up the problem correctly from the start.
In our exercise, differentiating $ u = 5x - 4 $ to find $ dx $ as $ \frac{du}{5} $ was a pivotal step that allowed us to rewrite the integral in a simpler form.

91影视

Exer. \(1-8\) : Evaluate the integral using the given substitution, and express the answer in terms of \(x\). $$ \int \frac{1}{(5 x-4)^{10}} d x ; \quad u=5 x-4 $$

Short Answer

Step by step solution

Identify the Substitution

Find du/dx and dx

Substitute Into the Integral

Integrate with Respect to u

Substitute Back in Terms of x

Key Concepts

Substitution Method

Definite Integral

Indefinite Integral

Derivative Differentiation

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