Home > Calculus > Integrals of Rational Functions

How do you integrate #(x^3+x^2+2x+1)/((x^2+1)(x^2+2)) dx#?

Answer 1

Charles Anctil

#(x^3+x^2+2x+1)/((x^2+1)(x^2+2)) = (x^3+2x)/((x^2+1)(x^2+2)) + (x^2+1)/((x^2+1)(x^2+2))#

# = x/(x^2+1)+1/(x^2+2)#

There terms may be integrated by substitution.

For the first, use #u = x^2+1# to get #1/2ln (x^2+1)#.

For the second use #x=sqrt2u# to get #1/sqrt2 tan^-1(x/sqrt2)#

We can write the integral as

#1/2ln(x^2+1)+sqrt2/2tan^-1(x/sqrt2)#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Evelyn Agard

To integrate ( \frac{x^3 + x^2 + 2x + 1}{(x^2 + 1)(x^2 + 2)} ) with respect to ( x ), we can use partial fraction decomposition. After decomposing the fraction into its partial fraction form, we can integrate each term separately.

First, we express the given fraction as:

[ \frac{x^3 + x^2 + 2x + 1}{(x^2 + 1)(x^2 + 2)} = \frac{Ax + B}{x^2 + 1} + \frac{Cx + D}{x^2 + 2} ]

After finding the values of ( A ), ( B ), ( C ), and ( D ), we integrate each term separately. Then, we add the integrals together to find the final result.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.