Dr. J's Maths.com
Where the techniques of Maths
are explained in simple terms.

Calculus - Integration - Reverse Chain Rule - Multi-function 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 = ex + 3.

2. (i) Prove that .

(ii) Hence prove

  3. . 4. Evaluate
  5. Show . 6.

The diagram above shows the curve y = loge x and the line y = loge 4.

(i) Solve the equation loge x = loge 4. Hence write down the coordinates of the point of intersection of the two functions.

(ii) Show that the area of the shaded region is 3 u2.

(iii) Find the exact value of .


The diagram shows the graphs of y = loge x and the tangent to

y = loge x at the point (e, 1).

(i) Show that the equation of the tangent is .

(ii) Explain why for all positive values of x:
except for x = e.

(iii) Hence show that .

Exponential and trig functions. 9.
Hint.Let u = e2x.
10. .
Hint.Let u = tan x.
Hint.Let u = sin x.

13. and simplify your answer fully.

Log and trig functions. 15.
Hint.Let u = ln (cos x).
16. (i) Differentiate loge (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.

(ii) Find the coordinates of point A.

(iii) Show that

(iv) Find the area of the region between the two curves from the origin to Point A.

Answer.(i) A is (π/3, √3)
(iii) Area = (1 - ln2) u2.
18. Evaluate .
Hint.Let u = ln sin x.
  19. 20. Evaluate