Step-by-Step Integration of (x^2+2)/(x+1)

Now we'll integrate each term separately from our partial fraction decomposition.Let's start with the first term. We integrate x minus 1 using basic rules.We can separate this into the integral of x minus the integral of 1.This gives us x squared over 2 minus x plus a constant C-1.Now for the second term. We need to integrate 3 divided by x plus 1.We can factor out the constant 3 from the integral.To integrate this, we use the standard formula that the integral of 1 over u equals the natural logarithm of the absolute value of u plus a constant.In our case, u equals x plus 1.Applying this formula gives us 3 times the natural logarithm of the absolute value of x plus 1, plus another constant C-2.Now we combine both results.We get x squared over 2, minus x, plus C-1, plus 3 times the natural logarithm of the absolute value of x plus 1, plus C-2.We can simplify by combining the constants C-1 and C-2 into a single constant C.Our final result is x squared over 2, minus x, plus 3 times the natural logarithm of the absolute value of x plus 1, plus the constant C.This step demonstrates how breaking down a complex fraction into simpler terms makes integration more manageable.This is our final integrated expression for the original function.To verify our integration result, we need to differentiate our answer and check if we get back the original integrand.Let's differentiate our result with respect to x.We'll apply the differentiation rules term by term.The derivative of x squared over 2 is x. The derivative of x is 1. The derivative of 3 times natural log of absolute x plus 1 is 3 divided by x plus 1. And the derivative of the constant C is zero.Let's simplify our expression to compare it with the original integrand.We'll find a common denominator by multiplying the first two terms by x plus 1 over x plus 1.Expanding the numerator gives us x squared plus x minus x minus 1 plus 3.Simplifying, we get x squared plus 2 over x plus 1.Comparing our result with the original integrand, we can see they match perfectly.Since the derivative of our answer matches the original integrand, we've verified our solution is correct.Therefore, the final answer to our integration problem is x squared over 2 minus x plus 3 times the natural log of absolute x plus 1, plus an arbitrary constant C.