Evaluate the definite integral

[purdue2016]

Homework Statement

Evaluate the definite integral from 0 to 18.
∫[x/(9+4x)^1/2]dx

Homework Equations

The Attempt at a Solution

I know to use u-substitution and I set u = (9+4x)^1/2. I can't figure out where I'm messing up because I keep ending up with very large numbers which are incorrect.

 

[haruspex]

My first move would be to simplify the surd with x = 9u/4. Seeing the (1+u)^-1/2 that results, I would then use a trig substitution for u. tan² or sinh² will eliminate the surd.

 

[Mark44]

Homework Statement

Evaluate the definite integral from 0 to 18.
∫[x/(9+4x)^1/2]dx

Homework Equations

The Attempt at a Solution

I know to use u-substitution and I set u = (9+4x)^1/2. I can't figure out where I'm messing up because I keep ending up with very large numbers which are incorrect.

Your substitution should work. Show us what you did.

When you make your substitution, make sure that you change everything. You shouldn't have x's and dx's in your integral after the substitution.

 

[purdue2016]

I finally got it using the substitution I mentioned. I think I was screwing up because I didn't change the limits of integration for the new variable. Thanks for the help.

 

[Mark44]

My first move would be to simplify the surd with x = 9u/4. Seeing the (1+u)^-1/2 that results, I would then use a trig substitution for u. tan² or sinh² will eliminate the surd.

If the radicand were the sum or difference of squares, I would take this approach, but in this case a much simpler approach will work.

 

[Mark44]

I finally got it using the substitution I mentioned. I think I was screwing up because I didn't change the limits of integration for the new variable. Thanks for the help.

You don't necessarily need to change the limits of integration. If, after you have your antiderivative (as a function of u), you can undo the substitution to get the equivalent form in terms of x. At that point just plug in the limits of integration.

Schematically it's like this:
$$ \int_a^b f(x) dx = \int_{x = a}^b g(u)du = G(u)\vert_{x = a}^b = F(x)\vert_{x = a}^b = F(b) - F(a)$$

If you decide not to change the limits of integration, it's helpful to note that they are values of x by adding "x = ..." in the lower limit.