64. Using a computer algebra system, it was determined that
∫x(x+1)^8 dx = (x^10)/10 + (8x^9)/9 + (7x^8)/2 + 8x^7 + (35x^6)/3 + (56x^5)/5 + 7x^4 + (8x^3)/3 + x^2/2 + C.
Use integration by substitution to evaluate ∫x(x+1)^8 dx.

Verified step by step guidance

1. Identify the integral to solve: \(\int x (x+1)^8 \, dx\).
2. Choose a substitution to simplify the integral. Let \(u = x + 1\), so that \(du = dx\) and \(x = u - 1\).
3. Rewrite the integral in terms of \(u\): replace \(x\) with \(u - 1\) and \(dx\) with \(du\), giving \(\int (u - 1) u^8 \, du\).
4. Simplify the integrand: \(\int (u^{9} - u^{8}) \, du\).
5. Integrate each term separately: \(\int u^{9} \, du - \int u^{8} \, du = \frac{u^{10}}{10} - \frac{u^{9}}{9} + C\). Finally, substitute back \(u = x + 1\) to express the answer in terms of \(x\).

Key Concepts

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves choosing a substitution u = g(x) that transforms the integral into a simpler form in terms of u, making it easier to integrate. After integrating with respect to u, substitute back to the original variable x.

Chain Rule in Reverse

Integration by substitution is essentially the reverse process of the chain rule in differentiation. Recognizing the inner function and its derivative within the integrand helps identify the correct substitution. This connection allows us to rewrite the integral in a form that matches a basic integral formula.

Polynomial Expansion and Integration

Understanding how to integrate polynomials term-by-term is essential after substitution or expansion. Each term of the polynomial is integrated using the power rule, which states ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ -1. This knowledge helps verify the correctness of the integral obtained by substitution.