Math Snap

STEP 1

1. We are given the integral: $\int\left(\frac{2 x}{x^{2}+1}+\frac{1}{x^{2}+1}-\frac{2}{x-1}+\frac{1}{(x-1)^{2}}\right) \, dx$.
2. We need to find the indefinite integral of the given expression.

STEP 2

1. Break down the integral into separate terms.
2. Integrate each term individually.
3. Combine the results to obtain the final solution.

STEP 3

Break down the integral into separate terms:
$$\int \left( \frac{2x}{x^2+1} \right) \, dx + \int \left( \frac{1}{x^2+1} \right) \, dx - \int \left( \frac{2}{x-1} \right) \, dx + \int \left( \frac{1}{(x-1)^2} \right) \, dx$$

STEP 4

Integrate the first term: $\int \left( \frac{2x}{x^2+1} \right) \, dx$.
Use the substitution $u = x^2 + 1$, $du = 2x \, dx$.
$$\int \frac{2x}{x^2+1} \, dx = \int \frac{1}{u} \, du = \ln |u| + C_1 = \ln |x^2 + 1| + C_1$$

STEP 5

Integrate the second term: $\int \left( \frac{1}{x^2+1} \right) \, dx$.
This is a standard integral:
$$\int \frac{1}{x^2+1} \, dx = \tan^{-1}(x) + C_2$$

STEP 6

Integrate the third term: $-\int \left( \frac{2}{x-1} \right) \, dx$.
This is a standard integral:
$$-\int \frac{2}{x-1} \, dx = -2 \ln |x-1| + C_3$$

STEP 7

Integrate the fourth term: $\int \left( \frac{1}{(x-1)^2} \right) \, dx$.
This is a standard integral:
$$\int \frac{1}{(x-1)^2} \, dx = -\frac{1}{x-1} + C_4$$

Combine the results:
$$\int \left(\frac{2x}{x^2+1} + \frac{1}{x^2+1} - \frac{2}{x-1} + \frac{1}{(x-1)^2}\right) \, dx = \ln |x^2 + 1| + \tan^{-1}(x) - 2 \ln |x-1| - \frac{1}{x-1} + C$$ where $C = C_1 + C_2 + C_3 + C_4$ is the constant of integration.
The solution to the integral is:
$$\boxed{\ln |x^2 + 1| + \tan^{-1}(x) - 2 \ln |x-1| - \frac{1}{x-1} + C}$$