Integral Example 31

We’re asked to evaluate the integral

$$ \int \frac{1}{(1+x^2) \cdot \arctan x} \ dx $$

We approach this using the substitution method.

Let’s start by computing the differential of the arctangent function:

$$ d ( \arctan x ) = \frac{1}{1+x^2} \ dx $$

Solving for $ dx $, we get:

$$ dx = (1+x^2) \ d ( \arctan x ) $$

We now substitute this expression for $ dx $ into the integral:

$$ \int \frac{1}{(1+x^2) \arctan x} \ dx $$

$$ \int \frac{1}{(1+x^2) \arctan x} \cdot (1+x^2) \ d ( \arctan x ) $$

The terms $ (1+x^2) $ cancel out, leaving us with:

$$ \int \frac{1}{\arctan x} \ d ( \arctan x ) $$

Now, set t = $\arctan(x)$, so the integral becomes:

$$ \int \frac{1}{t} \ dt $$

This is a standard integral with a well-known result:

$$ \int \frac{1}{t} \ dt = \log|t| + C $$

Substituting back $ t = \arctan(x) $, we get:

$$ \log|\arctan x| + C $$

Therefore, the solution to the original integral is:

$$ \int \frac{1}{(1+x^2) \arctan x} \ dx = \log|\arctan x| + C $$

And that completes the calculation.