definite int's & substitution: int (4x+4)(x^2+2x)^5 dx

[annajee]

Hi, can someone help me understand this:
When evaluating the integral:

int (from -1 to 1) (4x + 4)(x^2 + 2x)^5 dx

if I use substitution I get the answer 242/3
but if I multiply it out, to get: int (from -1 to 1) (4x^8 + 128x^6 + 4x^7 + 128x^5) dx, and then evaluate the integral I get 5111/63.

So apparanty multiplying it out doesn't work? I am wondering why that gives a different result than using substitution method. It seems to me like it should work because whether you have those factors or you have the whole thing multiplied out it is still the same value, right? So why does it not come out to be the same when the integral is evaluated? I guess maybe I just don't understand substitution. Anyway, I appreciate some help here.

Thanks.

Anna

 

[stapel]

if I use substitution I get the answer 242/3 but if I multiply it out,...I get 5111/63.

Unfortunately, since we cannot see your steps, there is no way to determine where you may be going wrong. Sorry!

Eliz.

 

[galactus]

That's because when you used your u substitution, you did not change the limits of integration. A common mistake when using substitution.

\(\displaystyle 2\int\frac{x+1}{(x^{2}+2x)^{5}}dx\)

If we use the sub \(\displaystyle u=x^{2}+2x, \;\ du=(2x+2)dx, \;\ \frac{du}{2}=(x+1)dx\)

we have to change our limits to -1 to 3.

Sub your old limits into \(\displaystyle u=x^{2}+2x\) to find the new ones like so:

\(\displaystyle (1)^{2}+2(1)=3, \;\ (-1)^{2}+2(-1)=-1\)

EDIT:Thanks to SK for pointing out my blunder. I used 2x+2 instead of 4x+4, but the idea is still the same.