Evaluate the Integral

Problem

(∫_^)(√(,4*x−1)*d(x))

Solution

1. Identify the integral as a candidate for usubstitution because the integrand is a composition of a square root and a linear function.

2. Substitute u=4*x−1 to simplify the expression.

3. Differentiate u with respect to x to find d(u)=4*d(x) which implies d(x)=1/4*d(u)

4. Rewrite the integral in terms of u

(∫_^)(√(,u)⋅1/4*d(u))

5. Factor out the constant 1/4 and express the square root as a power:

1/4*(∫_^)(u(1/2)*d(u))

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

1/4⋅(u(3/2))/(3/2)+C

7. Simplify the coefficient by multiplying 1/4 by 2/3

1/6*u(3/2)+C

8. Back-substitute u=4*x−1 to return to the original variable x

Final Answer

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

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