AntiDerivative

An indefinite Integral

Integration Tips

Convert the terms into a sum of more easily integrable terms
- Then integrate each term, which you can do because of Integral#Properties
Convert Trigonometry functions into the form $\sin (a + b x)$
- Works for even powers of trig functions
Perform Partial Fractions decomposition if you have a fraction with top degree bigger than bottom
- Helps because $\pm$ can split up an integral
Graph
Split up absolute values by roots, then evaluate the integrals separately
Remember to Absolute Value the inside of $\ln$

U-Substitution

We basically hack the chain rule
So set $u = g (x), \frac{d u}{d x} = g^{'} (x), d u = g^{'} (x) d x$
Replace something of the form $\int f^{'} (g (x)) g^{'} (x) d x$
Substituting $u$ gives $\int f^{'} (u) d u = f (u) + C = f (g (x)) + C$
You may need to take a constant outside the integral
For a definite integral, make sure to update the limits to $g (a), g (b)$
sometimes, if you get the expression for $u$ in the derivative, it's helpful to sub in $u$

Integration By Parts

We basically hack the product rule

\frac{ \mathrm{d}(uv) }{ \mathrm{d}x } & =u\frac{ \mathrm{d}v }{ \mathrm{d}x }+v\frac{ \mathrm{d}u }{ \mathrm{d}x } \
\int u\frac{ \mathrm{d}v }{ \mathrm{d}x } +v\frac{ \mathrm{d}u }{ \mathrm{d}x } , dx & =uv \
\boxed{\int u , dv =uv-\int v , du }
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- If you have $x^{n}$ times $\sin ax$ or $\cos ax$ or $e^{ax}$, choose $u=x^{n},dv=f(x)dx$ - Don't worry if you have $dv=dx$ - If you have $\ln x$, try choosing $u=\ln x$ - If you get the original integral after integrating by parts a bunch of times, move it all to one side ### Trig Substitution - replace $\sqrt{ a^{2}-b^{2}x^{2} }$ with $bx=a\sin\theta$ to get $\sqrt{ a^{2}(1-\sin ^{2}\theta) }=|a\cos\theta|=a\cos\theta$ for $-\frac{\pi}{2}\leq \theta \leq \frac{\pi}{2}$ - Replace $\sqrt{ b^{2}x^{2}+a^{2} }$ with $bx=a\tan\theta$ and get $a\sec\theta$ for $-\frac{\pi}{2}\leq \theta \leq \frac{\pi}{2}$ - Or $bx=a\sinh u$ - Replace $\sqrt{ b^{2}x^{2}-a^{2} }$ with $bx=a\sec\theta$ to get $a\tan \theta$ for $-\pi \leq \theta \leq -\frac{\pi}{2} \cup 0 \leq \theta \leq \frac{\pi}{2}$ - Or $bx=a\cosh u$ - We can have these restrictions since they match the restrictions on the square root - you can also replace $x+c$ - Remember to switch back to $c$ - Draw triangles to help with this - remember to change $dx$ to $d\theta$ - Remember to change limits of integration - You can also useHyperbolic Functionssubstitution, which can be easier