(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The method to use to integrate the function is up to us.

The choices are:

1) U-substitution

2)Integration by Parts

3)Trigonometric integrals

4)Trigonometric substitution

5)Partial fraction

2. Relevant equations

According to me, the best way to do it is to use Integration by Parts.

Here is the function:

[tex]\int[/tex]sin(ln(x))/x^{2}

3. The attempt at a solution

u=sin(lnx)

du=cos(lnx)/x dx

dv=1/x^{2}dx

v=-1/x

[tex]\int[/tex]sin(lnx)/x^{2}dx = -sin(lnx)/x - [tex]\int[/tex] -cos(lnx)/x^{2}dx

= -sin(lnx)/x + [tex]\int[/tex] cos(lnx)/x^{2}dx

= -sin(lnx)/x + (-cos(lnx)/x^{2}- [tex]\int[/tex] sin(lnx)/x dx

2 [tex]\int[/tex] sin(lnx)/x^{2}dx = -sin(lnx)/x - cos(lnx)/x^{2}dx

[tex]\int[/tex]sin(lnx)/x^{2}dx = 1/2(-sin(lnx)/x - cos(lnx)/x^{2})

According to Maple, the answer is 1/2((-2sin(lnx) - 2cos(1/2ln(x))^{2}+ 1 + 2sin(1/2ln(x))cos(1/2lnx))/x

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# Homework Help: Integration by Parts with sin and ln(x)

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