INTEGRATING POWERS OF COSINE FUNCTIONS

Problem 1 :

cos2 x dx

Solution :

Given, cos2 x dx=1 + cos 2x2 dx=12 1 + cos 2x dx= 12dx + cos 2x dx= 12x + sin 2x2 + C= 12x + 14 sin 2x + C

Problem 2 :

cos2 (2x) dx

Solution :

Given, cos2 (2x) dx=1 + cos (2 × 2x)2 dx= 1 + cos 4x2 dx=12 1 + cos 4x dx= 12dx + cos 4x dx= 12x + sin 4x4 + C= 12x + 18 sin 4x + C

Problem 3 :

cos3 x dx

Solution :

Given, cos3 x dx= cos2x cos x dx= 1 - sin2x cos x dxu = sin x, du = cos x dx= 1 - u2 du= 1 du - u2 du= u - u33 + C= sin x - 13sin3x + C

Problem 4 :

cos4 x dx

Solution :

Given, cos4 x dx= cos2 x2 dx=1 + cos 2x22 dx= (1 + cos 2x)222 dx= (1 + cos 2x)24 dx= 14 (1 + cos 2x)2 dx= 14 12 + (cos 2x)2 + 2(1)(cos 2x) dx= 14 1 +cos2 2x + 2 cos 2x dx= 14dx + cos2 2x dx + 2 cos 2x dx= 14x + x2 + sin 4x8 + sin 2x + C= 14x + 18x + 132sin 4x + 14sin 2x +C= 38x + 132sin 4x + 14sin 2x +C

Problem 5 :

cos5 x dx

Solution :

Given, cos5 x dx= cos4x cos x dx= 1 - sin2x2 cos x dxu = sin x, du = cos x dx= 1 - u22 du=12 + u22 - 2(1)u2 du= 1 + u4 - 2u2 du= 1 du + u4 du - 2u2 du= u + u55 - 2u33+ C= sin x + 15sin5x - 23sin3x + C

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