Integration and Accumulation of Change

$\frac{x^2}{2} - \frac{1}{2}\ln(x^2 + 1) + C$

$\frac{x^2}{2} + \frac{1}{2}\ln(x^2 + 1) + C$

$x^2 - \ln(x^2 + 1) + C$

$\frac{x^2}{2} - \ln(x^2 + 1) + C$

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

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

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

$$\int \frac{1}{x + 1} dx$$

The degree of the numerator is less than the degree of the denominator.

The degree of the numerator is equal to the degree of the denominator.

The denominator is a constant.

The numerator is a constant.

$x^2 + 2x + 5\ln|x - 1| + C$

$x^2 - 2x + 5\ln|x - 1| + C$

2x^2 + 2x + 5\ln|x - 1| + C

$x^2 + 2x - 5\ln|x - 1| + C$

$$\int \frac{x^3 + 1}{x^2 + 4x + 4} dx$$

$$\int \frac{x}{x^2 + 4x + 4} dx$$

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

$$\int \frac{1}{x^2 + 4x + 4} dx$$

$x^2 + 6x + 9$

$x^2 - 4$

$x^2 + 4x + 1$

$x^2$

$\frac{1}{2}\arctan(\frac{x+1}{2}) + C$

$\arctan(\frac{x+1}{2}) + C$

$\frac{1}{2}\arcsin(\frac{x+1}{2}) + C$

$\arcsin(\frac{x+1}{2}) + C$

$\arcsin(\frac{x-1}{3}) + C$

$\frac{1}{3}\arcsin(\frac{x-1}{3}) + C$

$\arctan(\frac{x-1}{3}) + C$

$\frac{1}{3}\arctan(\frac{x-1}{3}) + C$