Common integrals, with the derivation

Each page gives the antiderivative up front, then derives it, shows where the absolute values come from, and lists the mistakes students actually make. No paywall in front of the steps.

• $\int \frac{1}{x}\,dx = \ln|x| + C$derivation
• $\int \ln x\,dx = x\ln x - x + C$derivation
• $\int e^x\,dx = e^x + C$derivation
• $\int \sin x\,dx = -\cos x + C$derivation
• $\int \cos x\,dx = \sin x + C$derivation
• $\int \tan x\,dx = -\ln\left|\cos x\right| + C$derivation
• $\int \sec x\,dx = \ln\left|\sec x + \tan x\right| + C$derivation
• $\int \sec^2 x\,dx = \tan x + C$derivation
• $\int \csc x\,dx = -\ln\left|\csc x + \cot x\right| + C$derivation
• $\int \cot x\,dx = \ln\left|\sin x\right| + C$derivation
• $\int \frac{1}{1+x^2}\,dx = \arctan x + C$derivation
• $\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C$derivation
• $\int \sqrt{x}\,dx = \frac{2}{3}x^{3/2} + C$derivation
• $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$derivation
• $\int a^x\,dx = \frac{a^x}{\ln a} + C$derivation
• $\int \sin^2 x\,dx = \frac{x}{2} - \frac{\sin(2x)}{4} + C$derivation
• $\int \tan^2 x\,dx = \tan x - x + C$derivation
• $\int x e^{x}\,dx = (x - 1)e^{x} + C$derivation
• $\int e^{2x}\,dx = \frac{1}{2}e^{2x} + C$derivation

Every antiderivative on these pages is verified automatically: a test differentiates it and confirms the integrand comes back, so a wrong sign fails the build instead of reaching you. Looking for the other direction? Common derivatives.