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Chapter 7: Problem 27

Evaluate the following integrals. $$\int \frac{d x}{\left(1+4 x^{2}\right)^{3 / 2}}$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int \frac{d x}{\left(1+4 x^{2}\right)^{3 / 2}}$$ Answer: The value of the integral is given by $$\frac{1}{2}\left(\tan^{-1}\left(\frac{2x}{1}\right) + C\right)$$

Step by step solution

01

Trigonometric Substitution

Let's use the substitution \(\tan(\theta)=\frac{2x}{1}\). We now must find the derivative of this equation with respect to \(x\) and express \(dx\) in terms of \(d\theta\).
02

Find the Derivative

Differentiating the above equation, we get: $$\sec^{2}(\theta)d\theta=2dx$$ and hence $$ dx= \frac{1}{2}\sec^{2}(\theta)d\theta $$
03

Substitute the Trig Identity and \(dx\) into the integral

Now, substitute \(\tan(\theta)=\frac{2x}{1}\) in the integral and also replace \(dx\) with \(\frac{1}{2}\sec^2(\theta)d\theta\). Our integral becomes: $$\int \frac{1}{\left(1+4 \left(\frac{1}{2}\tan(\theta)\right)^{2}\right)^{3 / 2}}\frac{1}{2}\sec^{2}(\theta)d\theta$$ Simplify the expression inside the integral: $$\int \frac{1}{\left(1 + \tan^2(\theta)\right)^{3/2}} \frac{1}{2}\sec^2(\theta)d\theta$$ Now using the identity: \(\sec^2(\theta) = 1 + \tan^2(\theta)\), $$\int \frac{\frac{1}{2}\sec^2(\theta)}{\sec^3(\theta)} d\theta$$
04

Solve the Integral

After simplification, our integral becomes: $$\frac{1}{2}\int \sec^{-1}(\theta) d\theta$$ The integral of \(\sec^{-1}(\theta)\) is given by: $$\int \sec^{-1}(\theta)d\theta = \theta + C$$ So the result of the integral is: $$\frac{1}{2}\left(\theta + C \right)$$
05

Convert Back to an Expression in \(x\)

Now we need to re-write this result in terms of \(x\). We know that: $$\tan(\theta)=\frac{2x}{1}$$ Thus, we have the following relationship between \(\theta\) and \(x\): $$\theta = \tan^{-1}\left(\frac{2x}{1}\right)$$ Finally, substitute \(\theta\) back into the result of the integral: $$\frac{1}{2}\left(\tan^{-1}\left(\frac{2x}{1}\right) + C\right)$$ Hence, the evaluation of the integral is $$\int \frac{d x}{\left(1+4 x^{2}\right)^{3 / 2}} = \frac{1}{2}\left(\tan^{-1}\left(\frac{2x}{1}\right) + C\right)$$

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Most popular questions from this chapter

Suppose \(f\) is positive and its first two derivatives are continuous on \([a, b] .\) If \(f^{\prime \prime}\) is positive on \([a, b],\) then is a Trapezoid Rule estimate of \(\int_{a}^{b} f(x) d x\) an underestimate or overestimate of the integral? Justify your answer using Theorem 2 and an illustration.

Consider the curve \(y=\ln x\) a. Find the length of the curve from \(x=1\) to \(x=a\) and call it \(L(a) .\) (Hint: The change of variables \(u=\sqrt{x^{2}+1}\) allows evaluation by partial fractions.) b. Graph \(L(a)\) c. As \(a\) increases, \(L(a)\) increases as what power of \(a ?\)

Evaluate \(\int \frac{d y}{y(\sqrt{a}-\sqrt{y})},\) for \(a > 0\). (Hint: Use the substitution \(u=\sqrt{y}\) followed by partial fractions.)

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{\sqrt{1+\sqrt{x}}} ; x=\left(u^{2}-1\right)^{2}$$

a. Graph the functions \(f_{1}(x)=\sin ^{2} x\) and \(f_{2}(x)=\sin ^{2} 2 x\) on the interval \([0, \pi] .\) Find the area under these curves on \([0, \pi]\) b. Graph a few more of the functions \(f_{n}(x)=\sin ^{2} n x\) on the interval \([0, \pi],\) where \(n\) is a positive integer. Find the area under these curves on \([0, \pi] .\) Comment on your observations. c. Prove that \(\int_{0}^{\pi} \sin ^{2}(n x) d x\) has the same value for all positive integers \(n\) d. Does the conclusion of part (c) hold if sine is replaced by cosine? e. Repeat parts (a), (b), and (c) with \(\sin ^{2} x\) replaced by \(\sin ^{4} x\) Comment on your observations. f. Challenge problem: Show that, for \(m=1,2,3, \ldots\) $$\int_{0}^{\pi} \sin ^{2 m} x d x=\int_{0}^{\pi} \cos ^{2 m} x d x=\pi \cdot \frac{1 \cdot 3 \cdot 5 \cdots(2 m-1)}{2 \cdot 4 \cdot 6 \cdots 2 m}$$
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