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Chapter 12: Problem 1042

\((\pi / 4) \int_{0} \log (\cot 2 x)^{\sin 4 x} d x\) is equal to \(\ldots \ldots\) (a) 0 (b) \((\pi / 4)\) (c) \((\pi / 8)\) (d) \((\pi / 2)\)

Short Answer

Expert verified
The given integral, $(\pi / 4) \int_{0}^{\pi/4} \log (\cot 2 x)^{\sin 4 x} d x$, is equal to \(\boxed{0}\).

Step by step solution

01

Recognize the limits of integration

Notice that the integral limits are from 0 to \(\pi/4\). This fact will be helpful when working with trigonometric functions like cotangent and sine.
02

Simplify the integrand using properties of logarithms

We can simplify the integrand using logarithm properties. Recall that \(\log(a^b) = b \log(a)\). Therefore, the integrand can be rewritten as: \[ \sin(4x) \log(\cot(2x)). \] Now the integral becomes: \[ (\pi / 4) \int_{0}^{\pi/4} \sin(4x) \log(\cot(2x)) dx \]
03

Use substitution to deal with the cotangent function

Let \(u = \cot(2x)\). To find the corresponding differential, we will differentiate \(u\) with respect to \(x\): \[ \frac{du}{dx} = -2 \csc^2(2x). \] Now we can solve for \(dx\): \[ dx = -\frac{1}{2} \csc^2(2x) du. \] We also need to change the limits of integration. When \(x = 0\), we have \(u = \cot(0) = \infty\). When \(x = \pi/4\), we have \(u = \cot(\pi/2) = 0\). Now we can rewrite the integral in terms of \(u\): \[ (\pi / 4) \int_{\infty}^{0} -\frac{1}{2} \sin(4x) \log(u) \csc^2(2x) du. \]
04

Recognize the symmetry of the integrand

Notice that the sine function is an odd function, meaning \(\sin(-x) = -\sin(x)\). Cotangent and cosecant squared are even functions, meaning \(\cot(-x) = \cot(x)\) and \(\csc^2(-x) = \csc^2(x)\). Therefore, we can rewrite the integral as follows: \[ (\pi / 4) \int_{0}^{\infty} \frac{1}{2} \sin(-4x) \log(u) \csc^2(-2x) du. \] Now, we can use the fact that the sine function is odd and that cotangent and cosecant squared are even functions to rewrite the integral again: \[ -(\pi / 4) \int_{0}^{\infty} \frac{1}{2} \sin(4x) \log(u) \csc^2(2x) du. \]
05

Recognize that the integrand is zero

Since the original integral and the rewritten integral in steps 4 and 5 have equal values, we can write the following equation: \[ (\pi / 4) \int_{0}^{\pi/4} \sin(4x) \log(\cot(2x)) dx = -( \pi / 4) \int_{0}^{\infty} \frac{1}{2} \sin(4x) \log(u) \csc^2(2x) du. \] Adding these two integrals, we get: \[ 2 (\pi / 4) \int_{0}^{\pi/4} \sin(4x) \log(\cot(2x)) dx = 0. \] Finally, dividing by 2 and multiplying by \(\pi/4\), we get: \[ (\pi / 4) \int_{0}^{\pi/4} \log (\cot 2 x)^{\sin 4 x} d x = 0. \] Thus, the correct answer is (a) 0.

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