Antiderivative of Sin^3(x)Cos^4(x): Solutions & Examples

[tunabeast]

Homework Statement

Compute the following antiderivative $$\int (sin^3(x))(cos^4(x)) dx$$

Homework Equations

The Attempt at a Solution

If this problem requires use of integration by parts I'm struggling to work out to split it up and make it manageable. Have searched countlessly for a similar example on the net but have had no luck. Thanks

 

[HallsofIvy]

There is a standard "method" when you have a trig function to an odd power.

Since sin(x) it to the 3rd power, take one out to use with dx, convert sin²(x) to cos:
[tex]\int sin^3(x)cos^4(x)dx= \int sin^2(x)cos^4(x) sin(x)dx= \int (1- cos^2(x))cos^4(x) sin(x)dx[/itex]<br /> Now what substitution will make that easy?[/tex]

 

[fikus]

All you should do is new variable u = cos(x). You'll get integral u^4-u^6.

 

[colby2152]

Actually, $$u^6 - u^4$$ due to the negative in the derivative of $$cos(x)$$

 

[tunabeast]

fikus what do you mean, i can do a substitution straight away?

 

[rocomath]

fikus what do you mean, i can do a substitution straight away?

you can once you find that the derivative of your substitution appears in your original problem

 

[HallsofIvy]

fikus what do you mean, i can do a substitution straight away?

I believe that is what everyone as been trying to tell you!