\( \int \frac{\text { Integration using u-substitution }}{6(x+1)(x+2)^{7} d x} \)

To solve the integral \( \int \frac{1}{6(x+1)(x+2)^{7}} \, dx \) using u-substitution, we can set \( u = x + 2 \). Consequently, \( du = dx \) and \( x + 1 = u - 1 \). Transforming the integral gives us: \[ \int \frac{1}{6(u - 1)u^{7}} \, du \] This integral is now easier to solve. You can further simplify the integrand and then integrate via partial fractions or any suitable method. Want a little fun fact? U-substitution might seem like just a nifty trick for integrals, but it has roots in the chain rule of differentiation! It's a bridge connecting differentiation and integration, showcasing the harmony of calculus. If you’re interested in honing your skills, try exploring other forms of substitution, like trigonometric or exponential substitutions, to see how they play out in different types of integrals. Integrating becomes more like an exciting detective game when you unravel the right substitution!