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If I 1 =∫ sin -1 x dx and I 2 = ∫ sin -1 √1-x 2 dx then show that I 1 +I 2 = pie x / 2

If  I 1 =∫ sin -1 x dx    and I 2 = ∫ sin -1 √1-x 2  dx  then show that I 1 +I 2 = pie x / 2

Grade:12

1 Answers

Ramesh V
70 Points
14 years ago

we shld know that

arc.sin x +arc.sin y = arc.sin ( x.(1-y2)1/2 + y.(1-x2)1/2 )  for -1 < x,y < 1and x2+y2<=1

here put y = (1-x2)1/2  and it satisfies that for -1 < x < 1 , 0< (1-x2)1/2 < 1  

                                                                                           and  x2+1-x2=1

so I1 + I2 = integral ( arc.sin ( x2 + (1-x2)1/2.(1-x2)1/2 ))

                = integral ( arc.sin ( 1 ))

               = integral ( Pi/2 )

              = x.Pi/2 

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Naga Ramesh
IIT Kgp - 2005 batch

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