How to Calculate the Integral of sin(ln x)?

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In summary, the conversation is about the procedure for calculating the integral of sin(ln x). The suggested substitution x = e^z changes the integral to e^z sin(z) dz. The method for solving this integral is integration by parts, as explained in the abstract overview provided. The conversation also includes a reminder that the correct notation is ln x, not In x.
  • #1
kennis2
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Hi I hope you can help me because I got test in this Thursday :(

What´s the procedure to calculate Integral of sin(Inx)?

The result is: (x/2){[cos (ln x)] - [sin (ln x)]} + C

But I never get there…

Thanks,
 
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  • #2
Use the substitution x = e^z.
 
  • #3
can you so kind to do some procedures? because i don't get it :(
the teacher says this should be by parts..do u know how?
 
Last edited:
  • #4
Well making the substitution x = e^z changes the integral to the following:

[tex]\Int e^z Sin(z) dz [/tex]

The way to solve this integral is by working backwards from the product rule, which is called integration by parts. I'm sure that your calculus book has an example, here is an abstract overview of the technique:

http://mathworld.wolfram.com/IntegrationbyParts.html
 
  • #5
Two times part integration will do the transformed integral.It would have been much nicer

[tex] \int \ln\sin x \ dx [/tex]

Daniel.
 
  • #6
By the way, it is ln x NOT In x.

Am I the only person who gets really teed off by that?


(Sorry, but I have been seeing students writing "In" for "ln" for more years than I want to remember!)
 

FAQ: How to Calculate the Integral of sin(ln x)?

What is the integral of sin(Inx)?

The integral of sin(Inx) is -cos(Inx) + C, where C is a constant.

How do I solve the integral of sin(Inx)?

To solve the integral of sin(Inx), you can use the substitution method or integration by parts. For the substitution method, let u = Inx and du = 1/x dx. Then the integral becomes ∫sin(u)du = -cos(u) + C = -cos(Inx) + C. For integration by parts, let u = sin(Inx) and dv = dx, then du = Inx cos(Inx) dx and v = x. The integral becomes ∫sin(Inx)dx = -Inx cos(Inx) + ∫cos(Inx)dx. You can then use substitution or integration by parts again for the remaining integral.

What is the domain of the integral of sin(Inx)?

The domain of the integral of sin(Inx) is all real numbers except for x = 0, where the function is undefined.

Is there a shortcut for solving the integral of sin(Inx)?

Yes, there is a shortcut for solving the integral of sin(Inx) using the complex exponential function. The integral of sin(Inx) can be written as the imaginary part of e^(iInx). Then, using the power rule for integration, the integral becomes ∫sin(Inx)dx = Im(e^(iInx)) = Im(cos(Inx) + i sin(Inx)) = sin(Inx) + C.

Can I use trigonometric identities to solve the integral of sin(Inx)?

Yes, you can use trigonometric identities such as the double angle formula and half angle formula to simplify the integral of sin(Inx). This can make the integration process easier and faster.

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