Integrate Sine and Square root Composite Function

In summary, The conversation discusses a problem with integrating the function \int \frac{\sin^3(\sqrt{x})}{\sqrt{x}}dx and suggests using the identity \cos^2(x)+\sin^2(x)=1 \ \therefore \sin^2x=1- \cos^2(x) to simplify the integral. The idea of substituting u=\cos(x) is also mentioned, but the person struggles with the integration by parts method. The conversation ends with the suggestion to use the substitution \sqrt{x}=u to make the integral easier.
  • #1
Petrus
702
0
Hello MHB,
I got stuck on integrate this function

\(\displaystyle \int \frac{\sin^3(\sqrt{x})}{\sqrt{x}}dx\)
my first thinking was rewrite it as \(\displaystyle \int \frac{\sin^2(\sqrt{x})\sin(\sqrt{x})}{\sqrt{x}}dx\)
then use the identity \(\displaystyle \cos^2(x)+\sin^2(x)=1 \ \therefore \sin^2x=1- \cos^2(x)\)
\(\displaystyle \int \frac{(1-\cos^2(\sqrt{x}))\sin(\sqrt{x})}{\sqrt{x}}dx\)
subsitute \(\displaystyle u= \cos(x) \therefore du=- \sin(x) dx\)
then we get
\(\displaystyle - \int \frac{1-u^2}{ \cos^{-1}(u)}du\)
but that does not seem smart so my last ide is integrate by part, but I struggle on that part..

\(\displaystyle u= \sqrt{x}^{-1} \therefore du=\sqrt{x}^{-2}\) and \(\displaystyle dv=\sin^3(\sqrt{x}) \therefore v=?\)

Regards,
\(\displaystyle |\pi\rangle\)
 
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  • #2
Re: Integrate 2

Use the substitution $\sqrt{x}=u$
 
  • #3
Re: Integrate 2

ZaidAlyafey said:
Use the substitution $\sqrt{x}=u$
Hello Zaid,
I don't see what is the point with that=S? can I substitute twice or ? I am kinda clueless:confused:

Regards,
\(\displaystyle |\pi\rangle\)
 
  • #4
Re: Integrate 2

Petrus said:
Hello Zaid,
I don't see what is the point with that=S? can I substitute twice or ? I am kinda clueless:confused:

Regards,
\(\displaystyle |\pi\rangle\)

One thing to realize that \(\displaystyle (\sqrt{x})'=\frac{1}{2\sqrt{x}}\)

So before we proceed , we make \(\displaystyle u=\sqrt{x}\) substitution , which makes things easier since we are left with

\(\displaystyle 2\int \sin^3 (u)\,du \) which you can integrate , right ?

I have got to get some sleep now , if you are still stuck someone is always around :cool:.
 
  • #5
Re: Integrate 2

ZaidAlyafey said:
One thing to realize that \(\displaystyle (\sqrt{x})'=\frac{1}{2\sqrt{x}}\)

So before we proceed , we make \(\displaystyle u=\sqrt{x}\) substitution , which makes things easier since we are left with

\(\displaystyle 2\int \sin^3 (u)\,du \) which you can integrate , right ?

I have got to get some sleep now , if you are still stuck someone is always around :cool:.
Hello Zaid,
Now it make a lot sense!:) Thanks for taking your time and sleep well! I am also supposed to sleep but will do it soom =D

Regards,
\(\displaystyle |\pi\rangle\)
 

Related to Integrate Sine and Square root Composite Function

What is a composite function?

A composite function is a mathematical function that is made up of two or more functions. It can be written as f(g(x)), where f and g are individual functions. The output of g(x) is used as the input for f(x), creating a new function.

What is the process for integrating a sine and square root composite function?

To integrate a sine and square root composite function, we use the substitution method. We let u be equal to the square root function, and then we substitute u into the integral. This allows us to rewrite the integral in terms of u, which we can then integrate using the power rule.

Why is it important to learn how to integrate composite functions?

Integrating composite functions is important because many real-world problems involve multiple functions and require integration to solve. Being able to integrate composite functions allows us to model and solve a wide range of problems in fields such as physics, engineering, and economics.

What are some common mistakes when integrating a sine and square root composite function?

One common mistake is forgetting to use the chain rule when integrating the square root function. Another mistake is not substituting u back into the final answer, resulting in an incorrect solution. It is important to carefully follow the steps of the integration process to avoid these mistakes.

Can a composite function be integrated using other methods besides substitution?

Yes, there are other methods for integrating composite functions, such as integration by parts and trigonometric substitution. However, the substitution method is often the most efficient and straightforward method for integrating a sine and square root composite function.

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