Integral of (sin (x))^1/2)(cos^3(x)) dx

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Homework Help Overview

The discussion revolves around integrals involving trigonometric functions, specifically the integral of \((\sin(x))^{1/2} \cos^3(x) \, dx\) and \(\frac{\sin(\ln x)}{x} \, dx\). Participants are exploring methods to approach these integrals without providing complete solutions.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to begin the first integral. Another participant suggests rewriting the integral using trigonometric identities to facilitate the integration process. There is also a request for assistance with a second integral, indicating a struggle with starting points for such problems.

Discussion Status

The discussion is active, with participants offering suggestions and clarifications. One participant has provided a method for rewriting the first integral, while another seeks help with a different integral, indicating ongoing exploration of various approaches.

Contextual Notes

Participants are working within the constraints of homework assignments, which may limit the extent of guidance provided. There is an emphasis on understanding the setup and transformations of the integrals rather than arriving at final answers.

Jeann25
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Not really sure where to begin on this integral. Hope this reads alright on here.

Integral of (sin (x))^1/2)(cos^3(x)) dx
 
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\int {\sqrt{sin(x)}} cos(x)^3 dx

Convert this into

\int {\sqrt{sin(x)}}cos^2(x)cos(x).

From your identities in trignometry

cos^2(x) = 1 - sin^2(x).

This will leave you with all sine terms and one cosine term once you rewrite cosine squared. Do you see what to do from here?
 
Last edited:
That was very helpful, thank you. Could I also get help with this one pls? :) I just seem to have a hard time figuring out where to start on these.

The integral of (sin(ln x))/x dx
 
Jeann25 said:
That was very helpful, thank you. Could I also get help with this one pls? :) I just seem to have a hard time figuring out where to start on these.

The integral of (sin(ln x))/x dx

Use a substitution letting u = ln x.
 

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