INTEGRATING POWERS OF SINE FUNCTIONS

Problem 1 :

sin2 x dx

Solution :

Given, sin2 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 :

sin3 x dx

Solution :

Problem 3 :

sin4 x dx

Solution :

Given, sin4 x dx= sin2 x2 dx=1 - cos 2x22 dx= 14(1 - cos 2x)2 dx=1412 +( cos 2x)2 - 2(1)(cos 2x)dx= 141 + cos22x - 2 cos 2xdx= 141 +1 + cos 4x2 - 2 cos 2x dx= 14dx + 18(1 + cos 4x) dx - 24 cos 2x dx= 14dx + 18dx + 18cos 4x dx - 12cos 2x dx= x4 + x8 + 18 sin 4x4 - 12 sin 2x2 + C= 2x + x8 + 132 sin 4x - 14 sin 2x + C= 3x8 - 14 sin 2x + 132 sin 4x + C

Problem 4 :

sin5 x dx

Solution :

Given, sin5 x dx= sin4 x · sin x dx= sin2 x2 · sin x dx=1 - cos2 x2 · sin x dxu = cos x, du = -sin x dx= -1 - u22 ·du= - 12 + u4 - 2u2 du= - du - u4 du + 2u2 du= -u - u55 + 2u33 + C= -cos x - 15(cos x)5 + 23(cos x)3 + C= -cos x - 15cos5x + 23cos3x + C

Problem 5 :

tan3 x dx

Solution :

tan3x dx=sin3xcos3x dx=sin2xcos3xsin x dx=1-cos2xcos3xsin x dxu=cos x and du=-sinx dxsin x dx=-du=1-u2u3(-du)=1u3-1u(-du)=1u-1u3du= ln |u| - u-2-2+C= ln |u| + 12u2+C= ln |cosx| + 12cos2x+C= ln |cosx| + sec2x2+C

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