substituting a value into an integrand

Question

if the substitution √(x)=sin(y) is made into the integrand of (with the upper limit of: 1/2  and the lower limit of: 0) ∫ ( (√(x) )/( (√(1-x) )dx , the resulting integral is _______________.

 

 

Answer 1

when x is 0,sin y is 0 which implies y is 0

 

when x =1/2  implies sin y =1/√2  ⇒y= π/4  

 

so your limits are from 0 to π/4 

 

your integrand is siny /√1-sin²y

 

ie  siny/cos y 

ie tany  

                                                                     π/4

hence your integration problem converts to 0^{∫tan y}

 

now do it