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netineya [11]
1 year ago
6

Find the indefinite integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) dx x(l

n x2)5
Mathematics
1 answer:
Pepsi [2]1 year ago
7 0

I'm assuming the integral is

\displaystyle \int \frac{dx}{x (\ln(x^2))^5}

We have

\ln(x^2) = 2 \ln|x| \implies (\ln(x^2))^5 = 32 (\ln|x|)^5

Then substituting y=\ln|x| and dy=\frac{dx}x, the integral transforms and reduces to

\displaystyle \int \frac{dx}{x(\ln(x^2))^5} = \frac1{32} \int \frac{dy}{y^5} \\\\ ~~~~~~~~ = \frac1{32} \left(-\frac1{4y^4}\right) + C \\\\ ~~~~~~~~ = -\frac1{128(\ln|x|)^4} + C

which we can rewrite as

128 (\ln|x|)^4 = 8\cdot2^4(\ln|x|)^4 = 8 (2\ln|x|)^4 = 8 (\ln(x^2))^4

and so

\displaystyle \int \frac{dx}{x (\ln(x^2))^5} = \boxed{-\frac1{8(\ln(x^2))^4} + C}

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