I need to find the integral:

int:(x^2+2x+3)/(x^3+3x+9) dx

I think I am supposed to use the rule: int:1/u du=ln|u|+C

But I don't know what to do first.

Thanks for your help! :-)

Oh- and the answer the back of my book gives is (1/3)ln|x^3+3x^2+9x|+C

This is an integral in the form of

INT f'(x)/f(x) dx

it integrates to ln |(f(x))|

http://en.wikipedia.org/wiki/Table_of_integrals

Thanks for your help! :-)

I think I got the answer!

Thanks1

To find the integral of the given function, we can start by using partial fraction decomposition. However, before we do that, let's check if the denominator can be factored.

The denominator, x^3 + 3x + 9, doesn't have any simple factorizations like (x - a)(x - b)(x - c). So we'll proceed with partial fraction decomposition.

Let's assume that the integral can be written in the form:

∫ (x^2 + 2x + 3)/(x^3 + 3x + 9) dx = ∫ (A/x + Bx + C)/(x^3 + 3x + 9) dx

Now, we need to find the values of A, B, and C. To do this, we'll use a method called equating coefficients.

Multiplying both sides of the equation by (x^3 + 3x + 9), we get:

x^2 + 2x + 3 = A + Bx(x^3 + 3x + 9) + C(x^3 + 3x + 9)

Expanding the right side, we have:

x^2 + 2x + 3 = A + Bx^4 + 3Bx^2 + 9Bx + Cx^3 + 3Cx + 9C

Now, let's collect like terms. On the left side, we have:

x^2 + 2x + 3

On the right side, we group the terms with the same powers of x:

(A + 9C) + (Bx^4 + Cx^3) + (3Bx^2 + 3Cx)

Equate the coefficients of the corresponding powers of x on both sides:

For the constant term: A + 9C = 3
For the coefficient of x^4: B = 0
For the coefficient of x^3: C = 0
For the coefficient of x^2: 3B = 1
For the coefficient of x: 9B = 2

From the equations B = 0, C = 0, and A + 9C = 3, we find that A = 3 and C = 0.

Substituting these values back into 3B = 1, we get:

3B = 1
3(0) = 1
0 = 1 (which is not true)

There is no value of B that satisfies this equation. This usually means that there is a quadratic term in the denominator that cannot be factored further using real numbers. Consequently, we can't use the method of partial fraction decomposition in this case.

In order to find the integral, we can use a different method, such as trigonometric substitutions or integration by parts. However, these methods may result in a more complicated expression, which may not match the answer given in your book.

So, in this particular case, the integral may need to be evaluated using numerical methods or advanced techniques such as computer algebra systems.