Evaluate the Integral

Problem

(∫_^)(x2*(x3−1)4*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the derivative of the inner function x3−1 is a multiple of the outer factor x2

2. Define the substitution variable u=x3−1

3. Differentiate u with respect to x to find d(u)=3*x2*d(x) which implies x2*d(x)=1/3*d(u)

4. Substitute the expressions for u and d(x) into the integral to rewrite it in terms of u

(∫_^)(u4⋅1/3*d(u))

5. Factor out the constant 1/3 from the integral.

1/3*(∫_^)(u4*d(u))

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

1/3⋅(u5)/5+C

7. Simplify the resulting expression.

(u5)/15+C

8. Back-substitute the original expression x3−1 for u to get the final result in terms of x

Final Answer

(∫_^)(x2*(x3−1)4*d(x))=((x3−1)5)/15+C

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