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Question

Find the integral of the function tan(3ln(x))x

A
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B
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C
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D
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Solution

The correct option is D
We saw that tanxdx=ln|secx|+c
We have lnx and it’s derivative 1x in the function we want to integrate. So the substitution we make would be,
3 lnx = u.
(If we substitute lnx = u, we would need one more substitution to account for 3 inside tan function(or more steps).
3lnx=udu=3xdx1xdx=du3
So the integral tan(3ln(x))xdx becomes tan(u)du3
This will be equal to ln|sec(u)|3
Replacing u with 3 lnx, we get ln|sec(3lnx)|3
tan(3ln(x))xdx=ln|sec(3lnx)|3+c

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