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Evaluate the Integral integral of x(2x+5)^8 with respect to x

Problem

(∫_^)(x*(2*x+5)8*d(x))

Solution

  1. Identify the substitution method to simplify the integrand. Let u=2*x+5

  2. Differentiate the substitution equation to find d(u) Since u=2*x+5 then d(u)=2*d(x) which means d(x)=1/2*d(u)

  3. Solve for x in terms of u to replace the x outside the parentheses. From u=2*x+5 we get x=(u−5)/2

  4. Substitute these expressions into the integral.

(∫_^)((u−5)/2⋅u8⋅1/2*d(u))

  1. Factor out the constants and distribute u8 into the binomial.

1/4*(∫_^)((u9−5*u8)*d(u))

  1. Integrate term by term using the power rule.

1/4*((u10)/10−(5*u9)/9)+C

  1. Distribute the constant factor.

(u10)/40−(5*u9)/36+C

  1. Back-substitute u=2*x+5 to return to the original variable.

((2*x+5)10)/40−(5*(2*x+5)9)/36+C

Final Answer

(∫_^)(x*(2*x+5)8*d(x))=((2*x+5)10)/40−(5*(2*x+5)9)/36+C

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