Questions: (int frac11+x^2 dx)

$(int frac11+x^2 dx)$

Transcript text: $\int \frac{1}{1+x^{2}} dx$

Solution

Solution Steps

To solve the integral $\int \frac{1+x}{1+x^2} \, dx$, we can use the method of splitting the fraction into simpler parts. We can rewrite the integrand as $\frac{1}{1+x^2} + \frac{x}{1+x^2}$. The first part, $\frac{1}{1+x^2}$, is a standard integral that results in $\arctan(x)$. The second part, $\frac{x}{1+x^2}$, can be solved using a simple substitution.

Solution Approach

Split the integrand into two parts: $\frac{1}{1+x^2}$ and $\frac{x}{1+x^2}$.
Integrate each part separately.
Combine the results.

Step 1: Split the Integrand

We start by splitting the integrand $\frac{1+x}{1+x^2}$ into two simpler parts: \[ \frac{1+x}{1+x^2} = \frac{1}{1+x^2} + \frac{x}{1+x^2} \]

Step 2: Integrate Each Part Separately

Next, we integrate each part separately.

The integral of $\frac{1}{1+x^2}$ is: \[ \int \frac{1}{1+x^2} \, dx = \arctan(x) \]
The integral of $\frac{x}{1+x^2}$ can be solved using the substitution $u = 1 + x^2$, $du = 2x \, dx$: \[ \int \frac{x}{1+x^2} \, dx = \frac{1}{2} \int \frac{2x}{1+x^2} \, dx = \frac{1}{2} \ln|1+x^2| \]

Step 3: Combine the Results

Combining the results of the two integrals, we get: \[ \int \frac{1+x}{1+x^2} \, dx = \arctan(x) + \frac{1}{2} \ln|1+x^2| + C \]