Evaluate the Integral

Problem

(∫_^)(√(,1−4*y)*d(y))

Solution

1. Identify the integral as a composite function requiring usubstitution.

2. Substitute u=1−4*y

3. Differentiate u to find d(u)=−4*d(y) which implies d(y)=−1/4*d(u)

4. Rewrite the integral in terms of u

(∫_^)(√(,u)⋅(−1/4)*d(u))

5. Factor out the constant:

−1/4*(∫_^)(u(1/2)*d(u))

6. Apply the power rule for integration, (∫_^)(un*d(u))=(u(n+1))/(n+1)

−1/4⋅(u(3/2))/(3/2)+C

7. Simplify the coefficient:

−1/4⋅2/3*u(3/2)+C

−1/6*u(3/2)+C

8. Back-substitute u=1−4*y to get the final result.

Final Answer

(∫_^)(√(,1−4*y)*d(y))=−1/6*(1−4*y)(3/2)+C

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