How would I integrate the following:

(2x^2 + 5)/((x^2+1)(x^2+4))dx

I think I would start with making it a sum of two partial fractions.

To integrate the given rational function, you're correct that you need to express it as a sum of partial fractions first. Here's a step-by-step explanation of how to do that:

Step 1: Factor the denominator.
In this case, the denominator is (x^2+1)(x^2+4). Both these quadratic factors are irreducible, so we don't need to factor them further.

Step 2: Write the fraction as a sum of two partial fractions.
We'll express the given fraction as the sum of two partial fractions with unknown numerators, like this:

(2x^2 + 5) / ((x^2+1)(x^2+4)) = A / (x^2 + 1) + B / (x^2 + 4),

where A and B are the unknown coefficients to be determined.

Step 3: Find the common denominator.
To add the two partial fractions, you need a common denominator. In this case, it's (x^2 + 1)(x^2 + 4).

Step 4: Cross-multiply and equate coefficients.
Multiply both sides of the equation by the common denominator:

(2x^2 + 5) = A(x^2 + 4) + B(x^2 + 1).

By comparing the coefficients of the corresponding powers of x on both sides, you can solve for A and B. In this case, the coefficients of x^2 on each side would be:

2 = A + B,
and the constant terms would be:
5 = 4A + B.

Step 5: Solve the simultaneous equations.
Now, you have a system of two linear equations with two unknowns (A and B). Solve these equations simultaneously to find the values of A and B.

Upon solving, you'll find that A = 1/3 and B = 5/3.

Step 6: Rewrite the original integral using the partial fractions.
Finally, rewrite the original integral using the partial fractions:

∫ (2x^2 + 5) / ((x^2+1)(x^2+4)) dx = ∫ (1/3) / (x^2 + 1) dx + ∫ (5/3) / (x^2 + 4) dx.

Now, you can integrate each partial fraction separately. The integral of (1/3) / (x^2 + 1) can be found using the arctangent function, and the integral of (5/3) / (x^2 + 4) can be found using the inverse tangent function.