problems

• Jun 14th 2012, 06:11 AM
stampmagnet
problems

1. S((cosx(^3))sinx dx

2. S(x+1)sin((x^2)+2x dx

3. S(x^2)cos(x^3) dx

Can anyone explain me how I can apply the Gini index? Or use the Gini index when it comes down to calculus?

Thanks I appreciate it in advance.
• Jun 14th 2012, 06:25 AM
Reckoner
Re: problems
I guess 'S' is being used as an integral sign? All of these require only simple substitution:

Quote:

Originally Posted by stampmagnet
1. $\displaystyle \int\cos^3x\sin x\,dx$

Let $\displaystyle u = \cos x\Rightarrow du = -\sin x\,dx.$

Quote:

Originally Posted by stampmagnet
2. $\displaystyle \int(x+1)\sin\left(x^2+2x\right)\,dx$

Let $\displaystyle u = x^2 + 2x\Rightarrow du = 2x + 2\,dx.$

Quote:

Originally Posted by stampmagnet
3. $\displaystyle \int x^2\cos x^3\,dx$

Let $\displaystyle u = x^3\Rightarrow du = 3x^2\,dx.$
• Jun 14th 2012, 06:10 PM
stampmagnet
Re: problems
Yes the S is the integral. But how can I integrate the cos's and the sin's?
• Jun 14th 2012, 06:17 PM
Reckoner
Re: problems
Quote:

Originally Posted by stampmagnet
Yes the S is the integral. But how can I integrate the cos's and the sin's?

$\displaystyle \int\sin u\,du = -\cos u + C$

$\displaystyle \int\cos u\,du = \sin u + C$

The integration should be fairly straightforward after you make the substitutions. For example, take number 3. Using the substitution I stated earlier,

$\displaystyle \int x^2\cos x^3\,dx$

$\displaystyle =\frac13\int3x^2\cos x^3\,dx$

$\displaystyle =\frac13\int\cos u\,du$

$\displaystyle = \frac13\sin u + C$

$\displaystyle = \frac13\sin x^3 + C.$