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Chapter 7: Problem 11

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d x}{1-\cos 4 x}$$

Short Answer

Expert verified
$$\int \frac{dx}{1-\cos 4x} = -\frac{1}{4} \ln(\tan(2x)) + C$$

Step by step solution

01

Substitute to simplify

Let's first simplify the expression under the integral sign. We can do this by substituting: $$u = 4x$$ Then, we have that: $$du = 4dx$$ And so: $$dx = \frac{1}{4}du$$ Now, we can rewrite the integral as: $$\int \frac{dx}{1-\cos 4x} = \int \frac{1}{4}\frac{du}{1 - \cos u}$$
02

Apply the integral formula

Now, we can use a known integral formula to solve the integral we just obtained. In the table of integrals, we can find that: $$\int \frac{du}{1 - \cos u} = -\ln(\tan(\frac{u}{2})) + C$$
03

Substitute back

Now, we will substitute \(u = 4x\) back into the formula: $$- \frac{1}{4} \ln(\tan(\frac{4x}{2})) + C = -\frac{1}{4} \ln(\tan(2x)) + C$$
04

Write the final answer

We can now write the final answer for the integral: $$\int \frac{dx}{1-\cos 4x} = -\frac{1}{4} \ln(\tan(2x)) + C$$

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Most popular questions from this chapter

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{x^{4}+1}{x^{3}+9 x} d x$$

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \sqrt{e^{x}+1} d x \text { (Hint: Let } u=\sqrt{e^{x}+1}$$

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

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.

By reduction formula 4 in Section 3 $$\int \sec ^{3} u d u=\frac{1}{2}(\sec u \tan u+\ln |\sec u+\tan u|)+C$$ Graph the following functions and find the area under the curve on the given interval. $$f(x)=\left(x^{2}-25\right)^{1 / 2},[5,10]$$
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