substituting in an integral

Question

If the substitution √(X)=sin y is made in the integrand of:

(top of the integral sign is (1/2), bottom of the sign is 0) ∫ ( √(X) )/( √(1-X) )dx, the resulting integral is_____________.

 

A.) (top of the integral sign is (1/2), bottom of the sign is 1) 2∫( (sin²y)/(cosy) )dy

B.) (top of the integral sign is (1/2), bottom of the sign is 1) 2∫sin²ydy

C.) (top of the integral sign is (Π/4), bottom of the sign is 0) ∫sin²ydy

D.) (top of the integral sign is (Π/4), bottom of the sign is 0) 2∫sin²ydy
E.) (top of the integral sign is (Π/6), bottom of the sign is 0) 2∫sin²ydy

 

 

Answer 1

Answer is D

 

sqrt(x)=sin(y)

 

sqrt(x)=sin(y)

Differentiating the above

(1/2)((x)^(-1/2))dx=cos(y)dy

dx=2sqrt(x)cos(y)dy

 

Plugging in back

 

integral (sqrt(x)dx/sqrt(1-x))

 

integral (2(sin(y)^2)cos(y)dy/cos(y))

 

=2*integral (sin(y)^2)dy

 

Now, limits-----

when x= 0.   y=0

 

when x=1/2.      y=pi/4

So, the new limits are 0 to pi/4

 

So, the answer is D

Answer 2

D.

x=sin²y

dx=2sinycosydy

√x/√(1-x)=siny/cosy

integrand is (siny/cosy)2sinycosydy=2sin²ydy

at x=0 y=0 and at x=1/2 y=π/4

∫2sin²ydy from 0 to π/4