Standard Integrals

This is a collection of integrals which have very standard solutions.1 You by no means need to memorise these in the same way that you should know the derivative of tan⁡−1(x)\tan^{-1}(x) or the antiderivative of sec⁡(x)\sec(x), but you can go a lot faster if you can quote or jump to their solutions on the spot.

If your standard derivatives and antiderivatives are your atoms or keys, then these are your functional groups or chords. They often show up within more complex questions, perhaps after a substitution or simplification.

The best way to commit these to heart is not so much to memorise them as it is to naturally encounter them in the wild. Very naturally, the method will become part of you as you derive the solutions over and over. Even better, you learn to appreciate how these methods are applied and fit into integration, which helps you spot when and where to apply them.

These integrals being standard and very generic is part of the reason why you won’t find questions like them on Integrity.

I trust you can work out and verify these for yourself, so no answers have been provided.

Polynomial

∫1x dxint rac{1}{sqrt{x}} dx

Trigonometric

Based

∫sin⁡2x dx∫cos⁡2x dxegin{align*} & int sin^2{x} dx \ & int cos^2{x} dx end{align*}

Nontrivial

∫tan⁡2x dx∫sin⁡3x dx∫tan⁡3x dx∫sec⁡2xtan⁡x dxegin{align*} & int an^2{x} dx \ & int sin^3{x} dx \ & int an^3{x} dx \ & int sec^2{x} an{x} dx end{align*}

Chaos

∫sec⁡3x dxegin{align*} & int sec^3{x} dx end{align*}

Exponential

∫ln⁡x dxegin{align*} & int ln{x} dx end{align*}

Compound

∫xex dx∫xln⁡x dxegin{align*} & int xe^x dx \ & int x ln{x} dx end{align*}

General

∫(sin⁡x)(2n−1) dx∫(tan⁡x)(2n−1) dx∫(sec⁡x)n(tan⁡x) dxegin{align*} & int (sin{x})^{(2n-1)} dx \ & int ( an{x})^{(2n-1)} dx \ & int (sec{x})^n ( an{x}) dx end{align*}
  1. No chemistry joke, lmao.↩