calculus integrals?

Question

⌠
[(6x)/(x^4 + 1)]dx
⌡

evaluate the definite integral?? + C?

Answer 1

indefinite integral, since there are no limits

use u = x^2 so the integrand becomes 3 du / (u^2 + 1) the antiderivative is 3 arctan u + C, which equals 3 arctan x^2 + C

Answer 2

A definite integral is when you have specific limits of integration, and doesn't involve an arbitrary constant. Since there are no limits here, we evaluate the indefinite integral, which does use an arbitrary constant.

∫6x / (x^4 + 1) dx; let u = x^2, du = 2x dx
= ∫2du / (u^2 + 1)
You may recognise ∫du / (u^2 + 1) as being arctan u, giving a result of 2 arctan (x^2) + c, but if not, here's how to derive it.
Let u = tan θ, du = sec^2 θ dθ
= ∫2 sec^2 θ dθ / (tan^2 θ + 1)
= ∫2 sec^2 θ dθ / (sec^2 θ) [using the identity sec^2 θ = 1 + tan^2 θ]
= ∫2 dθ
= 2 θ + c
= 2 arctan u + c
= 2 arctan (x^2) + c.