Integrate: (2x^2+5)/((x^2+1)(x^2+4))
I came up with:
(tan^-1)(x)-(1/2)((tan^-1)(2/x)) but it keeps coming back the wrong answer even though I integrated correctly. Is there a way to simplify this answer, and if so, how?
Your answer is correct, but I think you are supposed to simpify the last term using the formula:
arctan(1/x) = pi/2 - arctan(x).
The pi/2 term can then be absorbed in the integration constant.
To integrate the given expression:
∫(2x^2+5)/((x^2+1)(x^2+4)) dx
You attempted to solve it and arrived at the correct result, which is:
∫(2x^2+5)/((x^2+1)(x^2+4)) dx = arctan(x) - (1/2)arctan(2/x) + C
Now, let's simplify this answer by using the formula you mentioned:
arctan(1/x) = π/2 - arctan(x)
Applying this formula to the second term, we get:
(1/2)arctan(2/x) = (1/2)(π/2 - arctan(x))
= π/4 - (1/2)arctan(x)
Now, we can rewrite the original solution as:
arctan(x) - (1/2)arctan(2/x) + C
= arctan(x) - (π/4 - (1/2)arctan(x)) + C
= arctan(x) + (1/2)arctan(x) - π/4 + C
= (3/2)arctan(x) - π/4 + C
Therefore, the simplified answer is:
∫(2x^2+5)/((x^2+1)(x^2+4)) dx = (3/2)arctan(x) - π/4 + C
Where C is the constant of integration.