Powers, roots and rational functions

Integral of a constant \int k\,dx = kx + C The integral of a constant k is kx + C, and the integral of 1 is x + C. See why, the area picture, definite integrals and the mistakes to avoid. Integral of x \int x\,dx = \frac{x^2}{2} + C The integral of x is x²/2 + C. See why with the reverse power rule and a triangle, plus definite integrals, the linear form ax + b and common mistakes. Integral of x² \int x^2\,dx = \frac{x^3}{3} + C The integral of x² is x³/3 + C. See the reverse power rule, a Riemann sum proof, definite integrals, the (ax + b)² form and the mistakes students make. Integral of √x \int \sqrt{x}\,dx = \frac{2}{3}x^{3/2} + C The integral of √x is (2/3)x^(3/2) + C. Rewrite the root as a power, check by differentiating, and see √(ax + b), 1/√x and definite integral examples. Integral of 1/x \int \frac{1}{x}\,dx = \ln\lvert x\rvert + C The integral of 1/x is ln|x| + C. See why the power rule fails here, where the absolute value comes from, and how to handle 1/(ax + b) and f'/f. Integral of 1/x² \int \frac{1}{x^2}\,dx = -\frac{1}{x} + C The integral of 1/x² is -1/x + C. Rewrite it as x^(-2), check by differentiating, and see improper integrals like 1 to infinity and the classic -1 to 1 trap. Integral of |x| \int \lvert x\rvert\,dx = \frac{x\lvert x\rvert}{2} + C The integral of |x| is x|x|/2 + C. See how to split at zero, why the answer is not |x²/2|, and how to evaluate definite integrals like |x - 1| from 0 to 4. Integral of x/(x² + 1) \int \frac{x}{x^2 + 1}\,dx = \frac{1}{2}\ln\left(x^2 + 1\right) + C The integral of x/(x² + 1) is ½ ln(x² + 1) + C. See the u-substitution step by step, why no absolute value is needed, and related integrals like 1/(x² + 1). Integral of 1/(x² - 1) \int \frac{1}{x^2 - 1}\,dx = \frac{1}{2}\ln\left\lvert\frac{x - 1}{x + 1}\right\rvert + C The integral of 1/(x² - 1) is ½ ln|(x - 1)/(x + 1)| + C. See the partial fraction steps, the general 1/(x² - a²) formula and worked definite integrals.