Find the Integral (x-1)(x^2-2x+5)^5

Problem

(∫_^)((x−1)*(x2−2*x+5)5*d(x))

Solution

1. Identify a potential substitution by looking for a function and its derivative within the integrand.

2. Define the variable u as the inner function of the power.

u=x2−2*x+5

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

d(u)/d(x)=2*x−2

4. Factor the expression for d(u) to match the remaining part of the integrand.

d(u)=2*(x−1)*d(x)

5. Rearrange the differential to solve for the terms present in the integral.

1/2*d(u)=(x−1)*d(x)

6. Substitute u and 1/2*d(u) into the original integral.

(∫_^)(u5⋅1/2*d(u))

7. Factor out the constant from the integral.

1/2*(∫_^)(u5*d(u))

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

1/2⋅(u6)/6+C

9. Simplify the resulting expression.

(u6)/12+C

10. Back-substitute the original expression for u to get the final answer in terms of x

((x2−2*x+5)6)/12+C

Final Answer

(∫_^)((x−1)*(x2−2*x+5)5*d(x))=((x2−2*x+5)6)/12+C

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