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Chapter 8: Problem 5

Evaluate each integral in Exercises \(1-36\) by using a substitution to reduce it to standard form. $$ \int_{0}^{1} \frac{16 x d x}{8 x^{2}+2} $$

Short Answer

Expert verified
The value of the integral is \( \ln 5 \).

Step by step solution

01

Identify the Substitution

To simplify the integral, look for a substitution that will make the expression inside the integral more manageable. Here, notice that the expression in the denominator is a function of \(x^2\). We can use the substitution: \( u = 8x^2 + 2 \). This choice is often done to simplify a quadratic expression.
02

Differentiate the Substitution

To use the substitution, differentiate \( u = 8x^2 + 2 \) with respect to \( x \):\[ \frac{du}{dx} = 16x \].Thus, \( du = 16x \, dx \). This matches exactly the numerator \( 16x \, dx \) in the integral.
03

Rewrite the Integral in Terms of \( u \)

Replace \( 16x \, dx \) with \( du \), and \( 8x^2 + 2 \) with \( u \):\[ \int_{x=0}^{x=1} \frac{16 x \, dx}{8 x^2+2} = \int_{u=8\cdot0^2+2}^{u=8\cdot1^2+2} \frac{du}{u} = \int_{2}^{10} \frac{du}{u} \].
04

Integrate the Function

Recognize that the integral \( \int \frac{du}{u} \) is a standard logarithmic form:\[ \int \frac{du}{u} = \ln |u| + C \].Since the limits of integration are present, calculate the definite integral.
05

Evaluate the Definite Integral

Substitute the limits of integration into the antiderivative:\[ \left[ \ln |u| \right]_{2}^{10} = \ln |10| - \ln |2| \].This simplifies to:\[ \ln \left( \frac{10}{2} \right) = \ln 5 \].
06

Confirm the Result

Check to make sure all steps are done correctly, and the change of variables aligns with the original integral's limits. The result of \( \ln 5 \) confirms that everything has been done correctly mathematically.

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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 technique in Integral Calculus used to simplify an integral, making it easier to evaluate. It involves replacing a complex part of the integrand with a simpler variable, usually denoted as \( u \). This allows you to work with a simpler integral in terms of \( u \) instead of the original variable.

Steps to apply the substitution method:
  • Identify a portion of the integrand that can be substituted. Look for expressions that form the derivative of another part of the integrand.
  • Substitute the identified part with a new variable \( u \), and find the differential \( du \) in terms of the original variable.
  • Rewrite the entire integral in terms of \( u \). This often transforms the integral into a standard or known form.
  • Integrate with respect to \( u \), find the antiderivative, and substitute back to the original variable if necessary.
In the given exercise, by using \( u = 8x^2 + 2 \), we simplify the expression under the integration, transforming it into a recognizable logarithmic form.
Definite Integral
A Definite Integral calculates the accumulated area under a curve from one point to another. Unlike indefinite integrals, definite integrals provide a specific number as an answer because they include upper and lower limits of integration.

To evaluate a definite integral:
  • Find the antiderivative of the integrand function.
  • Apply the limits of integration by substituting them into the antiderivative.
  • Calculate the definite value by subtracting the value at the lower limit from the value at the upper limit.
The definite integral of the exercise takes the expression \( \int_{0}^{1} \ rac{16x \, dx}{8x^2+2} \) and calculates it between the limits \( x=0 \) and \( x=1 \). Converting it into a function of \( u \) allows the straightforward computation of the integral from \( u = 2 \) to \( u = 10 \).
Logarithmic Integration
Logarithmic Integration occurs when the antiderivative of an expression is a logarithm, usually in the form of \( \int \frac{1}{u} \, du = \ln |u| + C \). It is particularly useful when dealing with integrands that result in a natural logarithm once integrated.

The standard form of logarithmic integration:
  • Recognize the integrand as a derivative of a logarithmic function.
  • Find the antiderivative, which is the logarithm of the absolute value of \( u \), \( \ln |u| \).
  • If evaluating a definite integral, substitute the upper and lower limits into the antiderivative to find the result.
In the exercise, after substituting and simplifying, the integral \( \int \frac{du}{u} \) results in \( \ln |u| \). By plugging in the limits \( u=2 \) and \( u=10 \), the definite integral is solved as \( \ln 5 \).
Change of Variables
Change of Variables is a fundamental concept in integration, assisting in transforming complex integrals into simpler forms. It involves substituting a part of the integrand with a new variable to reduce complexity and find a more straightforward path to the solution.

With a successful change of variables:
  • One can often turn a complicated integration problem into a standard form that is easier to evaluate.
  • This technique is widely applicable in solving both definite and indefinite integrals.
  • Choosing the correct substitution is key, as it directly influences the simplicity of the transformed integral.
In our example, the change of variables from \( x \) to \( u = 8x^2 + 2 \) transformed a complex rational integrand into a simple logarithmic integral form, \( \int \frac{du}{u} \), which was then easy to integrate.

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Most popular questions from this chapter

Use the integral tables to evaluate the integrals. \(\int \frac{1}{8} \sinh ^{5} 3 x d x\)

In Exercises \(89-92\) , use a CAS to explore the integrals for various values of \(p\) (include noninteger values). For what values of \(p\) does the integral converge? What is the value of the integral when it does converge? Plot the integrand for various values of \(p .\) $$ \int_{-\infty}^{\infty} x^{p} \ln |x| d x $$

Use a CAS to perform the integrations. Evaluate the integrals a. \(\int x \ln x d x \) b. \(\int x^{2} \ln x d x\) c. \(\int x^{3} \ln x d x\) d. What pattern do you see? Predict the formula for \(\int x^{4} \ln x d x\) and then see if you are correct by evaluating it with a CAS. e. What is the formula for \(\int x^{n} \ln x d x, n \geq 1 ?\) Check your answer using a CAS.

Find the area of the surface generated by revolving the curve \(y=\sqrt{x^{2}+2}, 0 \leq x \leq \sqrt{2},\) about the \(x\) -axis.

Evaluate the integrals in Exercises \(1-28\). $$ \int \frac{\sqrt{y^{2}-25}}{y^{3}} d y, \quad y>5 $$
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