Misc 19 - Chapter 7 Class 12 Integrals

Transcript

Misc 19 Integrate the function sin 1 cos 1 sin 1 + cos 1 , 0, 1 Let = sin 1 cos 1 sin 1 + cos 1 We can write as sin 1 cos 1 sin 1 + cos 1 = sin 1 2 1 2 = 1 2 sin 1 2 1 = 2 2 sin 1 2 = 2 2 sin 1 2 2 = 4 sin 1 1 Integrating . . . sin 1 cos 1 sin 1 + cos 1 = 4 sin 1 1 = 4 sin 1 = 4 sin 1 + 1 Hence I = 4 1 + 1 Solving 1 = sin 1 Put = = 2 Differentiating . . . = 2 1 = 2 = 2 Therefore sin 1 = sin 1 .2 =2 sin 1 . =2 sin 1 Hence we take First function :- = sin 1 Second function :- g = =2[ sin 1 sin 1 =2 sin 1 2 2 1 1 2 2 2 + =2 2 2 sin 1 2 1 2 2 1 2 + = 2 sin 1 2 1 2 + Solving We can write 2 1 2 = 2 + 1 1 1 2 = 1 2 1 1 2 = 1 2 1 2 1 1 2 = 1 2 1 1 2 Integrating . . . 2 1 2 dt = 1 2 1 1 2 = 1 2 2 1 1 2 2 = 2 1 2 2 + 1 2 2 sin 1 1 sin 1 1 = 2 1 2 + 1 2 sin 1 sin 1 = 2 1 2 1 2 sin 1 Hence we can write 1 = 2 sin 1 + 2 1 2 = 2 sin + 2 1 2 1 2 sin 1 Putting = 1 = 2 sin + 2 1 2 1 2 sin 1 = sin + 2 1 1 2 sin 1 Hence = 4 1 + C 1 = 4 1 + 2 1 1 2 1 + C 1 = 4 1 + 2 2 1 2 1 + C 1 = 4 1 + 2 2 2 1 + C 1 = 1 4 2 + 2 2 + C 1 = + +