1. Q2iiA8-integration

Hi:
I have this differentiation problem which am completely not sure about. I have done the question part way, but would like to know if am heading in the right direction before I take the next step.
thank you

2. That looks difficult - better substitute u = 7 + e^(2x)

3. You mean?

4. Just in case a picture helps...

... where

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

The general drift being...

Spoiler:

_________________________________________
Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!

5. So this leaves me with a quotient. Should i now use integration by parts?

6. Originally Posted by stealthmaths
Hi:
I have this differentiation problem which am completely not sure about. I have done the question part way, but would like to know if am heading in the right direction before I take the next step.
thank you

You could just observe that:

$\displaystyle \frac{d}{dx} \left[\frac{1}{3}(7+e^{2x})^{3/2}\right]=e^{2x}(7+e^{2x})^{1/2}$

then use the fundamental theorem of calculus to find the integral.

CB

7. hi:
so I have got 143.0 (3s.f.)