Evaluate the Integral integral of x(x^2-1)^4 with respect to x

Problem

(∫_^)(x*(x2−1)4*d(x))

Solution

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

2. Define the substitution variable u=x2−1

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

4. Substitute the expressions for u and d(x) into the integral.

(∫_^)(x*(x2−1)4*d(x))=(∫_^)(u4⋅1/2*d(u))

5. Factor out the constant 1/2 from the integral.

1/2*(∫_^)(u4*d(u))

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

1/2⋅(u5)/5+C

7. Simplify the resulting expression.

(u5)/10+C

8. Back-substitute the original expression x2−1 for u to get the final result.

((x2−1)5)/10+C

Final Answer

(∫_^)(x*(x2−1)4*d(x))=((x2−1)5)/10+C

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