Substitution
Let's say we want to integrate
in order obtain its primitve function
over some interval
:
∫f(x)dx=F(x)+C,x∈[a,b]
Now let's suppose we can identify a surjective function
that is continuous on
whose derivative on that interval is also continuous.
We can then write.
∫f(g(x))⋅g′(x)dx=∫f(g(x)) dg(x)=F(g(x))+C
In a nushell, we've substituted
with
and solved the integral in the usual way.
The tricky part is identifying
. Without it, it would not be possible to
Example:
Ok, the
based and redpilled way to do this is to notice from the definition of
in the OP is
df(x)=f′(x)dx
.
That means we could write
g′(x)dx
as
:
∫f(g(x))⋅g′(x)dx=∫f(g(x)) dg(x)
So a funny thing based engineers like to do (which makes mathfags seethe and cope) is this:
∫sin(x)cos(x)dx=∫sin(x) dsin(x)=2sin2(x)+C
Basically, treating
like
and using
∫xdx=2x2+C
.
The
cringe and bluepilled way of solving this, of course, is recognizing
sin(2x)=2sin(x)cos(x)
∫sin(x)cos(x)dx=21∫sin(2x)dx=4−cos(2x)+C=42sin2(x)−1+C=2sin2(x)+constant(−41+C)=2sin2(x)+C