# Thread: Intergration by substitution problem...

1. ## Intergration by substitution problem...

Okay, I generally have no problems, except when the derivative of the term that needs to be substituted is a constant, say, 1 and the coefficient of the $\displaystyle dx$ or whatever in the original question is a pronumeral, say, $\displaystyle 3x$.

Here is an example:

Find the integral between $\displaystyle x=-3$ and $\displaystyle x=-2$ where $\displaystyle y=x(3+x)^7$ using the substitution $\displaystyle u=3+x$

I have found that, by differentiation, $\displaystyle du=dx=1$

My problem is:
Once i get to this stage:

$\displaystyle \int_{-3}^{-2}$ $\displaystyle (3+x)^7$ $\displaystyle x dx$

$\displaystyle x=-2$ then $\displaystyle u=1$ and
$\displaystyle x=-3$ then $\displaystyle u=0$

and the $\displaystyle (3+x)^7$ then becomes $\displaystyle u^7$
but what about the $\displaystyle x dx$?
how do I get that into $\displaystyle du$?
if the differentiated $\displaystyle u$ had an $\displaystyle x$ in it, then fine, I could do that, but since it doesn't, I'm in a pickle.

And you can only have constant terms in front of the whole integral such as $\displaystyle 1/2$$\displaystyle \int_{-3}^{-2}......$

But what do I do for these types of integrals?

2. Hi Bartimaeus,

U are using the subsitution u=3+x, so x=u-3, and du=dx

So finally u get to integrade u^7 *(u-3)du = u^8 - 3u^7 du ...

3. First of all: du = dx = 1 is nonsense.

$\displaystyle \int x(3+x)^7~dx$

Substituting u = 3 + x, we see that:
x = u - 3
dx = du

There are no additional differentiated terms you need to multiply the original integrand with in order for you to change the variable of integration, so you can just change your terms and integrate.

$\displaystyle \int x(3+x)^7~du = \int (u-3)u^7~du = \int u^8 - 3u^7~du$

4. Hello,

Here is a way to do.

$\displaystyle u=x+3$

--> $\displaystyle \frac{du}{dx}=1 \implies {\color{blue}dx=du}$

So when you're changing the integral, change dx with its formula with respect to du.

Then, u=x+3 implies that x=u-3. So every x remaining in the integral will be substituted by this expression.

$\displaystyle \int_{-3}^{-2} x(x+3)^7 ~dx=\int_{-3}^{-2} ({\color{red}u-3}) \cdot u^7 ~{\color{blue}du}=\int_{-3}^{-2} u^8-3u^7 ~du$

Don't forget that the boundaries -3 and -2 apply to x and not to u.
If you want to keep them, you will have to substitute back, that is to say when you got the integral in terms of u, substitute all the u's by x+3.
Otherwise, you can change the boundaries so that they apply to u :
x=-3 ---> u=0
x=-2 ---> u=1
The new boundaries are 0 and 1.