Evaluate the Integral

Question

₀∫³ (x dx)/(x2 +1)

Answer 1

Let u = x² + 1 meaning du/dx = 2x or that dx = du/2x

Now substitute to get:

The "x's" cancel and you can bring the 1/2 out in front of the integral leaving you with (1/2)₀∫³1/u du and, since the integral of 1/u is ln(u), it becomes 1/2(ln(u) evaluated between 0 and 3. Now back substitute "x² + 1" wherever there is a "u" to get (1/2)(ln(x²+1) evaluated between 0 and 3 or:

1/2(ln(3²+1) - ln(0²+1)) = 1/2(ln(10)-ln(1)) = 1/2(ln(10) - 0) = 1/2ln(10)