Trigonometric substitution

[hazy dock]

I´m struggling a little with this topic and I´m thinking it's because I don´t really understand why I can use it. I'm really struggling to get why I can just set x = 3sin(theta), 2sin(theta), sin(theta) it just feels like I'm assigning a random value to x.

[serene tree]

!da2a

[abstract drum]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[serene tree]

it's one specific type of substitution, in essence no different than setting $u=x^2-1$ in an integral like $\int x\sqrt{x^2-1},dx$

[final shard]

elrichardo1337

[serene tree]

as for why we do them? if you see some sort of "common-trig-identity-like structure" in an integral a trig sub can simplify it greatly

take for example $\int\frac{dx}{1+x^2}$

[final shard]

elrichardo1337

[serene tree]

taking $u=1+x^2$ wouldn't be very helpful here, as since $du=2x,dx,$ we would need an extra factor of $x$ in the numerator that we don't have here

[final shard]

elrichardo1337

[serene tree]

but notice the denominator and how it resembles the trig identity $1+\tan^2\theta=\sec^2\theta$

[final shard]

elrichardo1337

[serene tree]

so that gives us the idea to set $x=\tan\theta,$ $dx=\sec^2\theta,d\theta$

[final shard]

elrichardo1337

[serene tree]

the integral in that case becomes $\int\frac{dx}{1+x^2}=\int\frac{\sec^2\theta,d\theta}{1+\tan^2\theta}=\int\frac{\sec^2\theta,d\theta}{\sec^2\theta}=\int d\theta=\theta+C$

[final shard]

elrichardo1337

[serene tree]

but what's $\theta$ in terms of $x?$ take the arctan of our original substitution to get $\theta=\arctan x,$ so indeed this integral is $\arctan x+C$ as expected

[final shard]

elrichardo1337

[serene tree]

does that help motivate it a bit better

[hazy dock]

I see that does make a lot more sense I completely miss understood what we were doing! Now I feel silly since substitution is literally in the name... Thank you so much for the help!

[serene tree]

npnp

it's usually taught separately from u sub bc the computations in trig subs tend to be more involved

but fundamentally yeah they're the same thing

[hazy dock]

.close