Evaluate the following integrals:

It is know that sin-1x+cos-1x = π/2

Tip – If f1(x) and f2(x) are two functions , then an integral of the form can be INTEGRATED BY PARTS as

where f1(x) and f2(x) are the first and second functions respectively.

Now, for the first term,

Taking f1(x) = sin-1√x and f2(x) = 1,

Taking (1-x)=a2,

Again, x=1-a2

Replacing the value of a, we get,

The total integration yields as

, where c’ is the integrating constant

For the second term,

Taking f1(x) = cos-1√x and f2(x) = 1,

Taking (1-x)=a2,

Again, x=1-a2

Replacing the value of a, we get,

The total integration yields as

, where c’’ is the integrating constant

where c is the integrating constant

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