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Chapter 4: Problem 52

Determine the following indefinite integrals. Check your work by differentiation. $$\int \frac{2}{16 z^{2}+25} d z$$

Short Answer

Expert verified
Answer: The indefinite integral of the function \(f(z) = \frac{2}{16z^{2} + 25}\) is given by \(\int f(z) dz = \frac{1}{2}\arctan{\left(\frac{4z}{5}\right)} + C\).

Step by step solution

01

Identify the substitution

Observe that the given integral has the form of: $$\int \frac{2}{16 z^{2}+25} d z$$ Let's choose the substitution: $$u = 4z$$ Now compute the derivative of u with respect to z: $$\frac{du}{dz} = 4$$
02

Rewrite the integral in terms of u

Now we can rewrite the integral in terms of u by substituting our variable. First, substitute u into the integral. We have: $$\int \frac{2}{(4z)^{2}+25} dz$$ Now replace \(4z\) with \(u\) and \(dz\) with the corresponding expression in \(du\). We get: $$\frac{1}{4}\int \frac{2}{u^{2}+25} du$$
03

Use trigonometric substitution

We can use the trigonometric substitution to solve the integral: $$u = 5 \tan{(\theta)}$$ Now compute the derivative of u with respect to \(\theta\): $$\frac{du}{d\theta} = 5 \sec^2{(\theta)}$$
04

Rewrite the integral in terms of u and 胃

Replace u with the trigonometric expression, and du with the corresponding expression in terms of d胃: $$\frac{1}{4}\int \frac{2}{(5 \tan{(\theta)})^{2}+25} (5 \sec^2{(\theta)}) d\theta$$
05

Simplify and solve the integral

Simplify the expression and solve the integral with respect to 胃: $$\frac{1}{4}\int \frac{10 \sec^2{(\theta)}}{25\tan^2{(\theta)} + 25} d\theta = \frac{1}{4}\int \frac{2 \sec^2{(\theta)}}{\tan^2{(\theta)} + 1} d\theta$$ We know that \(\tan^2{(\theta)} + 1 = \sec^2{(\theta)}\), so now the integral becomes: $$\frac{1}{4}\int 2 d\theta = \frac{1}{2}\theta + C$$
06

Transform back to z

Now we need to transform our solution back to the variable z. From our previous substitution, we have: $$u = 5 \tan{(\theta)}$$ $$\theta = \arctan{\left(\frac{u}{5}\right)}$$ Replace u with the expression in terms of z: $$\theta = \arctan{\left(\frac{4z}{5}\right)}$$ Now substitute this into our previous answer: $$\frac{1}{2}\arctan{\left(\frac{4z}{5}\right)} + C$$ Therefore, the indefinite integral of the given function is: $$\int \frac{2}{16z^{2} + 25} dz = \frac{1}{2}\arctan{\left(\frac{4z}{5}\right)} + C$$
07

Check solution by differentiation

Now we'll verify our solution by differentiating it with respect to z: $$\frac{d}{dz}\left[\frac{1}{2}\arctan{\left(\frac{4z}{5}\right)} + C\right] = \frac{1}{2}\frac{1}{1 + \left(\frac{4z}{5}\right)^2} \cdot \frac{4}{5} = \frac{2}{16z^{2} + 25}$$ Since the derivative of our solution matches the original function, our answer is correct: $$\int \frac{2}{16z^{2} + 25} dz = \frac{1}{2}\arctan{\left(\frac{4z}{5}\right)} + C$$

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Most popular questions from this chapter

Suppose \(f(x)=1 /(1+x)\) is to be approximated near \(x=0\). Find the linear approximation to \(f\) at 0 . Then complete the following table showing the errors in various approximations. Use a calculator to obtain the exact values. The percent error is \(100 \cdot |\) approximation \(-\) exact \(|/|\) exact \(| .\) Comment on the behavior of the errors as \(x\) approaches 0 .

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