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Chapter 5: Problem 40

Use a change of variables to evaluate the following definite integrals. $$\int_{0}^{2} \frac{2 x}{\left(x^{2}+1\right)^{2}} d x$$

Short Answer

Expert verified
Short Answer: The value of the definite integral is \(\frac{4}{5}\).

Step by step solution

01

Determine the substitution

Let's use the substitution \(u = x^2 + 1\). Then, we will need to find the derivative of \(u\) with respect to \(x\), which is \(\frac{du}{dx} = 2x\).
02

Change the limits of integration

Since the integral is definite, we must change the limits of integration with respect to the new variable \(u\). From the substitution, we have: When \(x = 0\), \(u = (0)^2 + 1 = 1\) When \(x = 2\), \(u = (2)^2 + 1 = 5\) So our new limits of integration are 1 to 5.
03

Substitute and simplify the integrand

With the substitution \(u = x^2 + 1\) and \(du = 2x\, dx\), our integral becomes: $$\int_{1}^{5} \frac{2 x}{(x^{2}+1)^{2}} d x = \int_{1}^{5} \frac{1}{u^2} \, du$$
04

Integrate with respect to u

Now, let's integrate the simplified expression: $$\int_{1}^{5} \frac{1}{u^2} \, du = - \frac{1}{u} \bigg\rvert_1^5$$
05

Evaluate the definite integral

Finally, let's evaluate the definite integral using the new limits of integration: $$- \frac{1}{u}\bigg\rvert_1^5 = - \frac{1}{5} + \frac{1}{1} = 1 - \frac{1}{5} = \frac{4}{5}$$ So, the value of the definite integral is \(\frac{4}{5}\).

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Most popular questions from this chapter

Use geometry to evaluate the following integrals. $$\int_{-2}^{3}|x+1| d x$$

Consider the function \(f(x)=x^{2}-4 x\) a. Graph \(f\) on the interval \(x \geq 0\) b. For what value of \(b>0\) is \(\int_{0}^{b} f(x) d x=0 ?\) c. In general, for the function \(f(x)=x^{2}-a x,\) where \(a>0,\) for what value of \(b>0\) (as a function of \(a\) ) is \(\int_{0}^{b} f(x) d x=0 ?\)

Consider the function g. which is given in terms of a definite integral with a variable upper limit. a. Graph the integrand. b. Calculate \(g^{\prime}(x)\) c. Graph g, showing all your work and reasoning. $$g(x)=\int_{0}^{x} \sin \left(\pi t^{2}\right) d t \text { (a Fresnel integral) }$$

Additional integrals Use a change of variables to evaluate the following integrals. $$\int\left(\sin ^{5} x+3 \sin ^{3} x-\sin x\right) \cos x d x$$

Evaluate \(\frac{d}{d x} \int_{-x}^{x}\left(t^{2}+t\right) d t\) Separate the integral into two pieces.)
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