1. 2.8 Integration of Trigonometric Functions
Lec 12 0f 12
Trigonometry
2.8 Integration of Trigonometric Functions

2. Learning Outcomes
c) find the integral of trigonometric functions by using integration by parts
d) evaluate definite integrals which involves trigonometric functions
e) Find the area of a region and the volume of the solid of revolution

3. The technique of Integration : Integration by parts
REMEMBER
The technique of Integration :
Integration by parts
The application of Integration :
finding the area of a region and the volume of a solid of revolution O

4. Using Integration by Parts to find the integral of trigonometric functions
EXAMPLE 1
Find
Solution
Let u = x2
So, du = 2x dx
and dv = sin x dx
v =
= - cos x

5. Let u = x du = dx and dv = cos x dx = x sin x - = x sin x + cos x
use the formula again
Let u = x
du = dx
and dv = cos x dx
= x sin x -
= x sin x + cos x

6. Thus

7. Use integration by parts formula twice Solution
EXAMPLE 2
Use integration by parts formula twice
Solution
Let u = e2x
So, du = 2e2x dx
and dv = sin x dx
v = - cos x
= - e2x cos x -
= - e2x cos x +
Use the formula again

8. Let u = e2x so, du = 2e2x dx and dv = cos x dx v = sin x = e2x sin x -
Hence
= - e2x cos x +
2 (e2x sin x -
= -e2x cos x + 2 e2x sin x - 4
Note : the integral is the same as the given question on the left

9. Thus
= -e2x cos x + 2 e2x sin x
( -e2x cos x + 2 e2x sin x) + c

10. Evaluating Definite Integrals which involves Trigonometric Functions
EXAMPLE 3
Evaluate

11. Substitute
u = sin 3x , so
du = 3cos 3x dx
1/3 du = cos 3x dx

13. and dv = cos2x dx => v = Let u = x => du = dx
Use integration by parts
and dv = cos2x dx => v =
Let u = x => du = dx

15. EXAMPLE 6
Prove that

16. Solution

18. Finding the area of a region and the volume of the solid of revolution
EXAMPLE 7
Find the area bounded by the curve y = x sinx and x-axis for x y
Solution
y = x sinx x

19. Area =
Use integration by parts
Let u = x and dv = sinx dx
du = dx , v = - cosx

20. EXAMPLE 8
The area enclosed the curve
and the line
is completely rotated about the line
Find the volume of the solid obtained.

21. Solution y
y = sin x x

24. TRY THIS !!!!
Sketch and find the area of the region bounded by the curve y = 1 + sinx, the lines x = 0 , x = and y = 0 .
The above region is rotated about the x - axis through 360o . Find the volume of the solid generated

25. y
Solution
y = 1 + sin x 2 1 x
Area of the shaded region

26. Volume =