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

Trigonometric functions - Calculus - Reverse Chain Rule.
Test Yourself 1.


The questions on this page all require you:

1. to identify a term to differentiate; and

2. to use the results of that differentiation to substitute for both the other term and the dx term;

3. then determine the integral in terms of that substituted variable (say u);

4. then substitute back into the original function.

No logs here. 1.
Hint.Let u = sin x.
Hint.Let u = cos x.
Hint.Let u = x2+1.
4. .
Hint.Change the tan and sec terms into sin and cos components. Then let u = cos 2x.
  5. .
Hint.Let n = cos x.
Hint.Let m = tan x.
Answer.Integral = 3√3.
Incorporates logs. 7. Find
Hint.Split the tan ratio into sin over cos. Then let u = cos x.
Answer.Integral = (1/2) ln2.
Hint.Let u = 1 - 2cos x.
Hint.Let u = tan x.
Hint.Let u = 1 + 2sin2x.
Miscellaneous 11. (i) Show that tan3 x can be written as tanx sec2 x - tan x.

(ii) Hence or otherwise show that

Hint.In part (ii), use a reverse chain rule on the two terms separately.
12. (i) Differentiate sin (x2).

(ii) Hence or otherwise, find the area bounded by y = xcos (x2), the x-axis and the lines x = 0 and x = 1.

(answer to 2 decimal places).
Answer.Area = 0.42 u2.