Evaluate the Integral

Problem

(∫_^)(√(,2*x−3)*d(x))

Solution

1. Identify the integral as a candidate for usubstitution because the integrand contains a linear function inside a square root.

2. Substitute u=2*x−3 to simplify the expression.

3. Differentiate u with respect to x to find d(u)=2*d(x) which implies d(x)=1/2*d(u)

4. Rewrite the integral in terms of u

(∫_^)(√(,u)⋅1/2*d(u))

5. Factor out the constant:

1/2*(∫_^)(u(1/2)*d(u))

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

1/2⋅(u(3/2))/(3/2)+C

7. Simplify the coefficients:

1/2⋅2/3*u(3/2)+C

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

8. Back-substitute u=2*x−3 to return to the original variable x

Final Answer

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

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