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Common Integrals

Intro​

This page lists several fundamental integrals and their basic properties. Their Derivations are mostly straightforward, as they follow directly from reversing those common derivatives.

Common Integrals​

Non-negotiable

The following integrals are essential formulas that should be memorized and known by heart, as they will be used frequently throughout the subsequent Derivations sections.

Common Integrals:
∫un du=un+1n+1+C(n≠−1)∫1u du=ln⁡∣u∣+C∫cos⁡u du=sin⁡u+C∫sin⁡u du=−cos⁡u+C∫tan⁡u du=ln⁡∣sec⁡u∣+C=−ln⁡∣cos⁡u∣+C∫cot⁡u du=ln⁡∣sin⁡u∣+C=−ln⁡∣csc⁡u∣+C∫sec⁡2u du=tan⁡u+C∫csc⁡2u du=−cot⁡u+C∫sec⁡utan⁡u du=sec⁡u+C∫csc⁡ucot⁡u du=−csc⁡u+C∫sec⁡u du=ln⁡∣sec⁡u+tan⁡u∣+C∫csc⁡u du=ln⁡∣csc⁡u−cot⁡u∣+C∫eu du=eu+C∫au du=auln⁡a+C(a>0, a≠1)∫11−u2 du=sin⁡−1u+C=arcsin⁡u+C∫11+u2 du=tan⁡−1u+C=arctan⁡u+C∫1uu2−1 du=sec⁡−1∣u∣+C=arcsec⁡∣u∣+C\begin{align} &\int{u^{n}}\ \mathrm{d}u=\frac{u^{n+1}}{n+1}+C \qquad (n\neq{-1})\\ &\int{\frac{1}{u}}\ \mathrm{d}u=\ln{\left|u\right|}+C\\ &\int{\cos{u}}\ \mathrm{d}u=\sin{u}+C\\ &\int{\sin{u}}\ \mathrm{d}u=-\cos{u}+C\\ &\int{\tan{u}}\ \mathrm{d}u=\ln{\left|\sec{u}\right|}+C=-\ln{\left|\cos{u}\right|}+C\\ &\int{\cot{u}}\ \mathrm{d}u=\ln{\left|\sin{u}\right|}+C=-\ln{\left|\csc{u}\right|}+C\\ &\int{\sec^2{u}}\ \mathrm{d}u=\tan{u}+C\\ &\int{\csc^2{u}}\ \mathrm{d}u=-\cot{u}+C\\ &\int{\sec{u}\tan{u}}\ \mathrm{d}u=\sec{u}+C\\ &\int{\csc{u}\cot{u}}\ \mathrm{d}u=-\csc{u}+C\\ &\int{\sec{u}}\ \mathrm{d}u=\ln{\left|\sec{u}+\tan{u}\right|}+C\\ &\int{\csc{u}}\ \mathrm{d}u=\ln{\left|\csc{u}-\cot{u}\right|}+C\\ &\int{e^u}\ \mathrm{d}u=e^u+C\\ &\int{a^u}\ \mathrm{d}u=\frac{a^u}{\ln{a}}+C \qquad (a>0,\ a\neq{1})\\ &\int{\frac{1}{\sqrt{1-u^2}}}\ \mathrm{d}u=\sin^{-1}{u}+C=\arcsin{u}+C\\ &\int{\frac{1}{1+u^2}}\ \mathrm{d}u=\tan^{-1}{u}+C=\arctan{u}+C\\ &\int{\frac{1}{u\sqrt{u^2-1}}}\ \mathrm{d}u=\sec^{-1}{\left|u\right|}+C=\operatorname{arcsec}{\left|u\right|}+C\\ \end{align}
u?

We use uu and du\mathrm{d}u instead of xx and dx\mathrm{d}x because uu may represent an entire function, not just a single variable. See more in Integral Techniques.