Evaluate the Integral

Problem

(∫_^)(9*z√(,3*z2−7)*d(z))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the expression inside the square root, 3*z2−7 is a multiple of the z term outside.

2. Define the substitution variable u=3*z2−7

3. Differentiate u with respect to z to find d(u)=6*z*d(z) which implies z*d(z)=1/6*d(u)

4. Substitute the expressions for u and d(z) into the original integral:

(∫_^)(9√(,u)⋅1/6*d(u))

5. Simplify the constant coefficients outside the integral:

9/6*(∫_^)(√(,u)*d(u))=3/2*(∫_^)(u(1/2)*d(u))

6. Integrate using the power rule (∫_^)(un*d(u))=(u(n+1))/(n+1)+C

3/2⋅(u(3/2))/(3/2)+C

7. Simplify the expression by canceling the 3/2 terms:

u(3/2)+C

8. Back-substitute the original expression 3*z2−7 for u to get the final result.

Final Answer

(∫_^)(9*z√(,3*z2−7)*d(z))=(3*z2−7)(3/2)+C

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