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Chapter 8: Problem 42

Use integration tables to find the integral. $$ \int_{0}^{4} \frac{2 x}{\sqrt{x^{2}+9}} d x $$

Short Answer

Expert verified
The integral of \( \int_{0}^{4} \frac{2 x}{\sqrt{x^{2}+9}} d x \) is 1.

Step by step solution

01

Identifying the problem

Identify the given integral as a form of \( \int \frac{x dx}{\sqrt{a^2 + x^2}} \). Here \( a=3 \) and the limits of integration are from 0 to 4.
02

Using substitution

Use the substitution \( u = a^2+b^2 \) which leads to \( du=2xdx \). The integral then changes to \( \frac{1}{2} \int \frac{du}{\sqrt{u}} \).
03

Solving the integral using integral table

Now use the formula from the integral table \( \int \frac{du}{\sqrt{u}} = 2\sqrt{u} + C \). So the answer is \( \sqrt{u} \).
04

Substitute Back

Substitute \( u \) back to \( x^2 + a^2 \) to get \( \frac{\sqrt{x^2+9}}{2} \). Then apply the limits of integration from 0 to 4 to get \( (\frac{\sqrt{4^2+9}}{2}) - (\frac{\sqrt{0+9}}{2}) \).
05

Final Calculation

Calculate the final value by plugging the limits into the final expression to get \( \frac{\sqrt{16+9}}{2} - \frac{\sqrt{9}}{2} = \frac{\sqrt{25}}{2} - \frac{\sqrt{9}}{2} = \frac{5}{2} - \frac{3}{2} = 1 \).

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Key Concepts

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

Integration Techniques
Integration techniques are methods used to solve integrals, which are fundamental in calculus. One common technique is substitution, which simplifies an integral by changing variables. This can make the integral easier to evaluate.

Another technique involves using integration tables. These tables contain a list of standard integral formulas that can be directly applied. This saves time and effort, especially with more complex integrals.

Let's consider the exercise:
  • Identify the form of the integral.
  • Decide on a suitable technique, such as substitution or using an integration table.
  • Apply these systematically to find the solution.
Understanding these techniques helps in tackling a wide range of integrals effectively.
Integration Tables
Integration tables are collections of formulas used to solve integrals quickly. They typically list standard integrals and their solutions, allowing you to find answers without detailed computation.

The given exercise used a formula from an integration table:
  • The integral of the form \( \int \frac{du}{\sqrt{u}} = 2\sqrt{u} + C \).
  • This helps us avoid complex algebraic manipulations.
These tables are especially useful when dealing with definite integrals over specific limits, as they allow us to focus on substituting and evaluating boundaries directly.

Thus, knowing how to utilize integration tables can make solving integrals much more efficient.
Substitution Method
The substitution method simplifies integrals by introducing a new variable. It is essentially reversing the chain rule of differentiation.

In our exercise:
  • We substituted \( u = x^2 + 9 \), making it easier to handle the integral.
  • This transformed the integral to \( \frac{1}{2} \int \frac{du}{\sqrt{u}} \).
  • By finding \( du = 2x dx \), it aligned perfectly with our integral allowing simplification.
After solving, substitute back to the original variable to apply the limits of integration.

Using substitution not only simplifies the integral but also demonstrates the interconnectedness of integration and differentiation. This method is crucial for tackling more challenging integrals effectively.

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Most popular questions from this chapter

Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b]\), then $$ f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t $$

True or False? , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of \(f\) is symmetric with respect to the origin or the \(y\) -axis, then \(\int_{0}^{\infty} f(x) d x\) converges if and only if \(\int_{-\infty}^{\infty} f(x) d x\) converges.

The velocity \(v\) of an object falling through a resisting medium such as air or water is given by $$ v=\frac{32}{k}\left(1-e^{-k t}+\frac{v_{0} k e^{-k t}}{32}\right) $$ where \(v_{0}\) is the initial velocity, \(t\) is the time in seconds, and \(k\) is the resistance constant of the medium. Use L'H么pital's Rule to find the formula for the velocity of a falling body in a vacuum by fixing \(v_{0}\) and \(t\) and letting \(k\) approach zero. (Assume that the downward direction is positive.)

State (if possible) the method or integration formula you would use to find the antiderivative. Explain why you chose that method or formula. Do not integrate. $$ \int x e^{x} d x $$

Surface Area Find the area of the surface formed by revolving the graph of \(y=2 \sqrt{x}\) on the interval \([0,9]\) about the \(x\) -axis.
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