Evaluate the Integral

Problem

(∫_^)((x2−9)3*(2*x)*d(x))

Solution

1. Identify the structure of the integrand and notice that the derivative of the inner function x2−9 is 2*x which is already present in the integral.

2. Substitute u=x2−9 to simplify the expression.

3. Differentiate u with respect to x to find d(u)=2*x*d(x)

4. Rewrite the integral in terms of u

(∫_^)(u3*d(u))

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

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

6. Simplify the resulting expression:

(u4)/4+C

7. Back-substitute the original expression x2−9 for u to get the final result.

Final Answer

(∫_^)((x2−9)3*(2*x)*d(x))=((x2−9)4)/4+C

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