Replace 
:
![\displaystyle \int_0^{\frac\pi2} \sqrt[3]{\tan(x)} \ln(\tan(x)) \, dx = \int_0^\infty \frac{\sqrt[3]{x} \ln(x)}{1+x^2} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_0%5E%7B%5Cfrac%5Cpi2%7D%20%5Csqrt%5B3%5D%7B%5Ctan%28x%29%7D%20%5Cln%28%5Ctan%28x%29%29%20%5C%2C%20dx%20%3D%20%5Cint_0%5E%5Cinfty%20%5Cfrac%7B%5Csqrt%5B3%5D%7Bx%7D%20%5Cln%28x%29%7D%7B1%2Bx%5E2%7D%20%5C%2C%20dx)
Split the integral at x = 1, and consider the latter one over [1, ∞) in which we replace 
:
![\displaystyle \int_1^\infty \frac{\sqrt[3]{x} \ln(x)}{1+x^2} \, dx = \int_0^1 \frac{\ln\left(\frac1x\right)}{\sqrt[3]{x} \left(1+\frac1{x^2}\right)} \frac{dx}{x^2} = - \int_0^1 \frac{\ln(x)}{\sqrt[3]{x} (1+x^2)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_1%5E%5Cinfty%20%5Cfrac%7B%5Csqrt%5B3%5D%7Bx%7D%20%5Cln%28x%29%7D%7B1%2Bx%5E2%7D%20%5C%2C%20dx%20%3D%20%5Cint_0%5E1%20%5Cfrac%7B%5Cln%5Cleft%28%5Cfrac1x%5Cright%29%7D%7B%5Csqrt%5B3%5D%7Bx%7D%20%5Cleft%281%2B%5Cfrac1%7Bx%5E2%7D%5Cright%29%7D%20%5Cfrac%7Bdx%7D%7Bx%5E2%7D%20%3D%20-%20%5Cint_0%5E1%20%5Cfrac%7B%5Cln%28x%29%7D%7B%5Csqrt%5B3%5D%7Bx%7D%20%281%2Bx%5E2%29%7D%20%5C%2C%20dx)
Then the original integral is equivalent to
![\displaystyle \int_0^1 \frac{\ln(x)}{1+x^2} \left(\sqrt[3]{x} - \frac1{\sqrt[3]{x}}\right) \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_0%5E1%20%5Cfrac%7B%5Cln%28x%29%7D%7B1%2Bx%5E2%7D%20%5Cleft%28%5Csqrt%5B3%5D%7Bx%7D%20-%20%5Cfrac1%7B%5Csqrt%5B3%5D%7Bx%7D%7D%5Cright%29%20%5C%2C%20dx)
Recall that for |x| < 1,
so that we can expand the integrand, then interchange the sum and integral to get
Integrate by parts, with
Recall the Fourier series we used in an earlier question [27217075]; if 
where 0 ≤ x ≤ 1 is a periodic function, then
Evaluate f and its Fourier expansion at x = 1/2 :
So, we conclude that
![\displaystyle \int_0^{\frac\pi2} \sqrt[3]{\tan(x)} \ln(\tan(x)) \, dx = \frac94 \times \frac{2\pi^2}{27} = \boxed{\frac{\pi^2}6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_0%5E%7B%5Cfrac%5Cpi2%7D%20%5Csqrt%5B3%5D%7B%5Ctan%28x%29%7D%20%5Cln%28%5Ctan%28x%29%29%20%5C%2C%20dx%20%3D%20%5Cfrac94%20%5Ctimes%20%5Cfrac%7B2%5Cpi%5E2%7D%7B27%7D%20%3D%20%5Cboxed%7B%5Cfrac%7B%5Cpi%5E2%7D6%7D)
I believe it could be weight today=193 goal weight=182
The value of y when x is equal to zero will be 330.
<h3>What is a system of equations?</h3>
A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
The given function is;
y = 330 + 25x
We have to find the value of y when x is equal to zero.
So,
Put x = 0
y = 330 + 25x
y = 330 + 25(0)
y = 330
Hence, the value of y when x is equal to zero will be 330.
Learn more about equations here;
brainly.com/question/10413253
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Answer: Pipes
Step-by-step explanation: Pipes transfer water around and veins transfer blood around the body
