Use a substitution to change the integral into one that can be found in an integral table. Then evaluate the integral. \( \int \frac{3 x^{3}+3 x+11}{\left(x^{2}+1\right)^{2}} d x \)

To evaluate the integral \[ \int \frac{3x^{3} + 3x + 11}{(x^{2} + 1)^{2}} \, dx, \] we can use substitution and break it down into simpler parts. ### Step 1: Substitution Let \( u = x^{2} + 1 \). Then, the derivative of \( u \) with respect to \( x \) is: \[ \frac{du}{dx} = 2x \quad \Rightarrow \quad du = 2x \, dx \quad \Rightarrow \quad x \, dx = \frac{du}{2}. \] ### Step 2: Rewrite the Integral Express the numerator \( 3x^{3} + 3x + 11 \) in terms of \( u \): \[ 3x^{3} = 3x(x^{2}) = 3x(u - 1). \] Now, the integral becomes: \[ \int \frac{3x(u - 1) + 3x + 11}{u^{2}} \, dx = \int \frac{3xu - 3x + 3x + 11}{u^{2}} \, dx = \int \left( \frac{3x}{u} + \frac{11}{u^{2}} \right) \, dx. \] ### Step 3: Split the Integral Separate the integral into two parts: \[ \int \frac{3x}{u} \, dx + \int \frac{11}{u^{2}} \, dx. \] ### Step 4: Evaluate Each Part 1. **First Integral:** \[ \int \frac{3x}{u} \, dx = \frac{3}{2} \int \frac{du}{u} = \frac{3}{2} \ln|u| + C_1 = \frac{3}{2} \ln(x^{2} + 1) + C_1. \] 2. **Second Integral:** \[ \int \frac{11}{u^{2}} \, dx = 11 \int \frac{1}{(x^{2} + 1)^{2}} \, dx. \] A standard integral formula gives: \[ \int \frac{1}{(x^{2} + 1)^{2}} \, dx = \frac{x}{2(x^{2} + 1)} + \frac{1}{2} \arctan(x) + C_2. \] Therefore: \[ 11 \left( \frac{x}{2(x^{2} + 1)} + \frac{1}{2} \arctan(x) \right) + C_2. \] ### Step 5: Combine the Results Putting it all together: \[ \int \frac{3x^{3} + 3x + 11}{(x^{2} + 1)^{2}} \, dx = \frac{3}{2} \ln(x^{2} + 1) + \frac{11x}{2(x^{2} + 1)} + \frac{11}{2} \arctan(x) + C, \] where \( C = C_1 + C_2 \) is the constant of integration. **Final Answer:** \[ \boxed{\,\frac{3}{2}\ln(x^{2}+1)\;+\;\frac{11\,x}{2(x^{2}+1)}\;+\;\frac{11}{2}\tan^{-1}(x)\,+\,C\,} \]