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. |
2.
Hint.Let u = cos x. |
3.
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. |
6.
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. |
8.
Hint.Let u = 1 - 2cos x. |
9.
Hint.Let u = tan x. |
10.
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. |