Question

integral of (x+1)/(3x^2)

Answer 1

Integral ( (1 + 3x)/(1 + x^2) dx )

To solve this, split into two fractions and two integrals.

Integral ( 1/(1 + x^2) dx) + Integral ( (3x)/(1 + x^2) dx )

The first integral is easy, because it's a known derivative.
It's the derivative of arctan(x).

arctan(x) + Integral ( (3x)/(1 + x^2) dx )

The second integral can be solved using substitution.

arctan(x) + 3 * Integral ( x/(1 + x^2) dx)
arctan(x) + 3 * Integral ( 1/(1 + x^2) x dx)

Let u = 1 + x^2. Then
du = 2x dx, so
(1/2) du = x dx

arctan(x) + 3 * Integral ( 1/u (1/2) du )
arctan(x) + 3(1/2) Integral ( 1/u du )
arctan(x) + (3/2) Integral ( 1/u du )
arctan(x) + (3/2) ln|u| + C

But u = 1 + x^2, so back-substituting gives us

arctan(x) + (3/2) ln|1 + x^2| + C

and 1 + x^2 is always positive anyway, so there's no need for the absolute value sign.

arctan(x) + (3/2) ln(1 + x^2) + C

Answer 2

Thankyou Very much. The response was very quick and very helpful.

Answer 3

Your welcome! Glad I could help!