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Chapter 4: Problem 108

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{2+x^{2}}{1+x^{2}} d x$$

Short Answer

Expert verified
Question: Evaluate the indefinite integral of the given function: $$\int \frac{2+x^2}{1+x^2} dx$$ Answer: The indefinite integral of the given function is: $$(x + \arctan(x)) + C$$

Step by step solution

01

Rewrite the integrand as a sum

To simplify the integrand, let's write it as the sum of two terms: $$\int \frac{2+x^{2}}{1+x^{2}} dx = \int \left( \frac{2}{1+x^{2}} + \frac{x^2}{1+x^2} \right) dx$$
02

Integrate each term separately

Notice that the second term is easy to integrate as it is already in a simple form. The first term can be recognized as the derivative of \(\arctan(x)\). Now, integrate each term separately: $$\int \frac{2}{1+x^{2}} dx = 2\int \frac{1}{1+x^{2}} dx = 2\arctan(x) + C_1$$ $$\int \frac{x^2}{1+x^2} dx = \int \left( 1 - \frac{1}{1+x^2} \right) dx = x - \arctan(x) + C_2$$
03

Combine the results and find the general solution

Now, we can combine the results to get the general solution: $$\int \frac{2+x^2}{1+x^2} dx = 2\arctan(x) + x - \arctan(x) + C$$ Where \(C = C_1 + C_2\) is a constant representing the constant of integration. So, the indefinite integral is: $$\int \frac{2+x^2}{1+x^2} dx = (x + \arctan(x)) + C$$
04

Check the result by differentiation

To verify our result, let's find the derivative of our solution: $$\frac{d}{dx} (x + \arctan(x)) = 1 + \frac{1}{1 + x^2}$$ Notice that this expression is slightly different from the integrand. Therefore, we need to further simplify the integrand: $$\frac{2+x^2}{1+x^2} = \frac{1}{1+x^2} + \frac{x^2 + 1}{1+x^2} = \frac{1}{1+x^2} + 1.$$ Now, comparing this expression with our derivative, we can see they match, confirming that the integral evaluation is correct.

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Most popular questions from this chapter

Show that \(f(x)=\log _{a} x\) and \(g(x)=\) \(\log _{b} x,\) where \(a>1\) and \(b>1,\) grow at a comparable rate as \(x \rightarrow \infty\)

Is it possible? Determine whether the following properties can be satisfied by a function that is continuous on \((-\infty, \infty)\). If such a function is possible, provide an example or a sketch of the function. If such a function is not possible, explain why. a. A function \(f\) is concave down and positive everywhere. b. A function \(f\) is increasing and concave down everywhere. c. A function \(f\) has exactly two local extrema and three inflection points. d. A function \(f\) has exactly four zeros and two local extrema.

Interpreting the derivative The graph of \(f^{\prime}\) on the interval [-3,2] is shown in the figure. a. On what interval(s) is \(f\) increasing? Decreasing? b. Find the critical points of \(f .\) Which critical points correspond to local maxima? Local minima? Neither? c. At what point(s) does \(f\) have an inflection point? d. On what interval(s) is \(f\) concave up? Concave down? e. Sketch the graph of \(f^{\prime \prime}\) f. Sketch one possible graph of \(f\)

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$f(x)=2 x^{4}+8 x^{3}+12 x^{2}-x-2$$

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime \prime}(x)=672 x^{5}+24 x, F^{\prime \prime}(0)=0, F^{\prime}(0)=2, F(0)=1$$
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