Questions: Evaluate the integral. (Use symbolic notation and fractions where needed. Use C for the constant of integration.) [ int frac110(x+1)(x^2+9)^2 dx= ]

Solution Steps

To evaluate the integral, we can use partial fraction decomposition to break down the integrand into simpler fractions that are easier to integrate. Once decomposed, we integrate each term separately and combine the results, adding the constant of integration \( C \) at the end.

We start with the integral to evaluate: \[ \int \frac{110}{(x+1)(x^{2}+9)^{2}} \, dx \]

Using partial fraction decomposition, we can express the integrand as a sum of simpler fractions. This allows us to integrate each term separately.

After performing the integration, we find: \[ \int \frac{110}{(x+1)(x^{2}+9)^{2}} \, dx = \frac{110(x + 9)}{180x^{2} + 1620} + \frac{11 \log(x + 1)}{10} - \frac{11 \log(x^{2} + 9)}{20} + \frac{77 \tan^{-1}\left(\frac{x}{3}\right)}{135} + C \]

Final Answer