Evaluate the Integral integral of x(8x+7)^8 with respect to x

Problem

(∫_^)(x*(8*x+7)8*d(x))

Solution

1. Identify the method of integration by substitution to simplify the expression.

2. Substitute u=8*x+7 which implies d(u)=8*d(x) or d(x)=1/8*d(u)

3. Solve for x in terms of u to replace the remaining x in the integrand: x=(u−7)/8

4. Rewrite the integral in terms of u

(∫_^)((u−7)/8⋅u8⋅1/8*d(u))

5. Factor out the constants from the integral:

1/64*(∫_^)((u−7)*u8*d(u))

6. Distribute u8 into the binomial:

1/64*(∫_^)((u9−7*u8)*d(u))

7. Integrate term by term using the power rule:

1/64*((u10)/10−(7*u9)/9)+C

8. Distribute the constant 1/64 into the terms:

(u10)/640−(7*u9)/576+C

9. Back-substitute u=8*x+7 to return to the original variable x

((8*x+7)10)/640−(7*(8*x+7)9)/576+C

Final Answer

(∫_^)(x*(8*x+7)8*d(x))=((8*x+7)10)/640−(7*(8*x+7)9)/576+C

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