integrat ((x^3+x^2+x+1)/(under root (x^2 + 2x + 3))

Question

∫((x³+x²+x+1)/(√x²+2x+3))

Answer 1

Rewrite the integral using factoring and complete the square.

 

(x²(x + 1) + x + 1) / √((x + 1)² + 2)) dx

 

 

Next, you can break up the integral as a sum of integrals to make it easier to work with.

 

          x²(x + 1)                                           x + 1

∫  _________________ dx       +   ∫   __________________  dx

       √[(x + 1)² + 2)]                            √[(x + 1)² + 2)]

 

 

Now you can use substitution.

 

Let        u = (x + 1)²                                  u^1/2 = x + 1

             du = 2(x + 1)dx                          -1 + √u = x

             (1/2)du = (x + 1)dx                    1 - 2u^1/2 + u = x²

 

               

Then, substitute back into original integral.

Answer 2

Factor it like this:

(x+1)(x^2 +1)
-------------------------
square-root( (x+1)^2 +2)



Let u = x+1

Then du = dx and
x = u-1

x^2 = u^2 -2u +1

x^2 +1 = u^2 - 2u +2

Now it is:

(u^2 - 2u +2) u
--------------------
square-root (u^2 +2)

rad2 = square-root(2)

Let u = rad2*tanT
Then du = rad2*secT^2 dT

This will cause the square
root in the denominator to drop out.

You will get three smaller integrals:

1) tanT*sinT
2) sinT
3) secT

The second is -cosT

The first can be integrated by parts
You can also use the table of integrals
for them.