Final answer:
To evaluate the given integral, use the method of partial fractions. Expand (x²+9)² and let u = x²+9. Take the derivative of u and substitute the values into the integral. Integrate u² and substitute u back in terms of x.
Step-by-step explanation:
To evaluate the integral ∫ 90((x+1)(x²+9)²) dx, we can use the method of partial fractions. First, expand (x²+9)² to x⁴+18x²+81. Then, let u = x²+9. Taking the derivative of u, we have du = 2x dx. Substituting these values into the integral, we get ∫ 90((x+1)(x²+9)²) dx = ∫ 90(u²) du. Integrating u² with respect to u gives (1/3)u³. Finally, substituting u back in terms of x gives (1/3)(x²+9)³+C, where C is the constant of integration.