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

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

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

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

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

Thus

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

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

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

Evaluating Definite Integrals which involves Trigonometric Functions EXAMPLE 3 Evaluate

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

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

EXAMPLE 6 Prove that

Solution

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

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

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

Solution y y = sin x x

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

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

Volume =