# Solving Integral: sin^3x cos^2x dx

• Ravenatic20
In summary: Then stop expecting people to do the problem for you. What has been suggested is that you rewrite the integral as $$\int sin^2(x)cos^2(x) (sin(x)) dx= \int (1- cos^2(x))cos^2(x) (sin(x)dx[/itex] Now, if u= cos(x), what is du? If you don't know that you should review differentiation before trying integration. Ravenatic20 I've been working on this problem for almost 4 days now and have made no progress. Once I think I've got it right, by professor says its wrong, and to try again. I've tried and tried. Any ideas? Here it is: [tex]\int sin^{3}x cos^{2}x dx$$

You could write everything in the integral in terms of sines, then either:
1. Use reduction formulae for sin^n x
2. write the powers of sines in terms of sines of multiples of x

Try to write the integrand as sin(x) * f(cos(x)) or cos(x) * f(sin(x)), where f is some algebraic function.

Ravenatic20 said:
I've been working on this problem for almost 4 days now and have made no progress. Once I think I've got it right, by professor says its wrong, and to try again. I've tried and tried. Any ideas? Here it is:

$$\int sin^{3}x cos^{2}x dx$$

I know this is not so formal, but it works:
using the fact that:

$$d(cos(x))=-sin(x)dx$$

you get:

$$\int sin^{3}(x) cos^{2}(x) dx=-\int sin^2(x)cos^2(x)d(cos(x))=-\int(1-cos^2(x))cos^2(x)d(cos(x))$$

and you are done.

regards
marco

Thanks marco but I need the whole integral solved so the $$\int$$ sign is removed. And to the point where our constant C is added on: $$+ C$$.

Ravenatic20 said:
Thanks marco but I need the whole integral solved so the $$\int$$ sign is removed. And to the point where our constant C is added on: $$+ C$$.

Do you honestly expect others to just do your work for you. Marco made the problem much simpler for you all it requires now is a simple, and fairly obvious substitution.

d_leet said:
Do you honestly expect others to just do your work for you. Marco made the problem much simpler for you all it requires now is a simple, and fairly obvious substitution.
No, the last part just doesn't make sense. If someone could explain it I'll take a shot at it, but I've never seen it (d(cos(x)))

Last edited:
Ravenatic20 said:
No, the last part just doesn't make sense. If someone could explain it I'll take a shot at it, but I've never seen it (d(cos(x))

do the substitution:

t=cos(x)-----> dt=d(cos(x))
you get it?

regards
marco

Thanks Marco.

This is what I have so far, in continuation of what Marco helped out with:
$$=-\int(1-cos^2(x))cos^2(x)d(cos(x))$$
$$=-\int(\frac{1}{2}-\frac{1}{2} cos2x)(\frac{1}{2}+\frac{1}{2} cos2x)d(cos(x))$$
$$=-[{(\frac{1}{2}x-\frac{1}{4} sin2x)(\frac{1}{2}x+\frac{1}{4} sin2x)] + C$$

Err, is this right? If not how do I fix it? Thanks

No, that's not right.

Do what macro_84 said. let t = cos(x)

Now substitute t everywhere you see a cos(x) in the integrand that macro gave you (the one with nothing but cosines). it's staring you in the face.

Err... That did not make much sense, sorry.

Ravenatic20 said:
Err... That did not make much sense, sorry.

How did that not make sense? Are you familiar with integration by substitution? Make the substititution t=cos(x) and what happens?

tx = -sin(x)?

Well no wonder you couldn't get this integral, you don't understand the most basic method of integration.

Vid said:
Well no wonder you couldn't get this integral, you don't understand the most basic method of integration.
Sorry, I've only been doing this for a few weeks. I came here for help, nothing else.

$$...=-\int(1-cos^2(x))cos^2(x)d(cos(x))=-\int(1-t^2)t^2dt$$

can you do it now??

ciao
marco

Then stop expecting people to do the problem for you. What has been suggested is that you rewrite the integral as
[tex]\int sin^2(x)cos^2(x) (sin(x)) dx= \int (1- cos^2(x))cos^2(x) (sin(x)dx[/itex]
Now, if u= cos(x), what is du? If you don't know that you should review differentiation before trying integration.

## What is the process for solving an integral with trigonometric functions?

The process for solving an integral with trigonometric functions involves using trigonometric identities and integration techniques to simplify the integral into a form that can be easily evaluated.

## What is the trigonometric identity used to solve this integral?

The trigonometric identity used to solve this integral is the double angle identity, specifically the identity sin(2x) = 2sin(x)cos(x).

## What integration technique should be used to solve this integral?

The integration technique that should be used to solve this integral is substitution. By substituting u = sin(x), the integral can be simplified into a form that can be easily evaluated.

## What are the limits of integration for this integral?

The limits of integration for this integral depend on the specific problem being solved. Typically, the limits will be given in terms of x, but may also be given in terms of u if substitution is used.

## What are the steps for solving this integral using substitution?

The steps for solving this integral using substitution are:

1. Identify a suitable substitution, typically involving one of the trigonometric functions present in the integral.
2. Apply the substitution to the integral, replacing the variable of integration with the new variable.
3. Simplify the integral using the substitution and any trigonometric identities.
4. Evaluate the integral using standard integration techniques.
5. Substitute the original variable back into the final answer.

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