Evaluate the integral: I = integral dx/((1+x²) arctan x)

Evaluate the integral: $$ I = \int \frac{dx}{(1+x^2)\arctan x}. $$

Explanation of Method

Since $\dfrac{d}{dx}(\arctan x)=\dfrac{1}{1+x^2}$, that exact factor multiplies the integrand alongside $1/\arctan x$. Substituting $u=\arctan x$ turns the whole integral into the standard logarithmic form $\int \frac{du}{u}$.

Substitute $u=\arctan x$.
Rewrite the integral in terms of $u$.
Apply the standard logarithmic antiderivative.
Substitute back and add the constant of integration.

Step 1 · Set up the substitution

Let:

$$ u = \arctan x. $$

Then:

$$ du = \frac{dx}{1+x^2}. $$

Step 2 · Rewrite the integral

Substituting into the integral:

$$ I = \int \frac{du}{u}. $$

Step 3 · Integrate

This is a standard logarithmic integral:

$$ I = \ln|u| + C. $$

Step 4 · Substitute back

Substituting back $u=\arctan x$:

$$ I = \ln|\arctan x| + C. $$

Summary Table

Quantity	Value
Substitution	$u = \arctan x$
$dx/(1+x^2)$ in terms of $du$	$du$
Antiderivative	$\ln\|u\|$

Additional Notes

The domain excludes $x=0$, since $\arctan0=0$ makes the integrand undefined there.
This is the prototype for the family of problems that follow ($108$–$109$): whenever $\arctan x$ (or a power of it) sits alongside its exact derivative $1/(1+x^2)$, substituting $u=\arctan x$ converts the problem into a plain algebraic integral in $u$.
The result can be checked by differentiating $\ln|\arctan x|$ with the chain rule and confirming it returns $\dfrac{1}{(1+x^2)\arctan x}$.

Result

The solution to the integral is:

$$ I = \ln|\arctan x| + C. $$