Calculus  Integration  Reverse Chain Rule  Multifunction expressions.
Test Yourself 1.
Before you start the serious stuff, read the following to show how logs and exponentials can combine to make you happy!!!!
Now to the serious stuff:
Exponential and log functions.  1. .
Hint.Let u = e^{x} + 3. 
2. (i) Prove that . (ii) Hence prove 
3. .  4. Evaluate
Answer.63/128 

5. Show .  6.  
7.
The diagram above shows the curve y = log_{e} x and the line y = log_{e} 4.


8.
The diagram shows the graphs of y = log_{e} x and the tangent to y = log_{e} x at the point (e, 1).


Exponential and trig functions.  9.
Hint.Let u = e^{2x}. 
10. . 
11.
Hint.Let u = tan x. 
12.
Hint.Let u = sin x. 

13. and simplify your answer fully. 
14.  
Log and trig functions.  15.
Hint.Let u = ln (cos x). 
16. (i) Differentiate log_{e} (cosx) with respect to x.
(ii) Hence or otherwise show 
17. (i) Sketch the graphs of y = 2 sinx and y = tan x for 0 ≤ x ≤ π/2. Mark A as the point of intersection of the graphs where x ≠ 0. Answer.(i) A is (π/3, √3) (iii) Area = (1  ln2) u^{2}. 
18. Evaluate .
Hint.Let u = ln sin x. 

19.  20. Evaluate 