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Techniques of Integration
The reverse chain rule, integration by substitution, integration by parts, additional strategies
The reverse chain rule, integration by substitution, integration by parts, additional strategies
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If F(x)=∫f(x)dx, then
Integrating a composition of functions f(g(x)) requires us to divide by g′(x), so it is easier to find the anti-derivative of anything of the form g′(x)f′(g(x)) by first dividing by g′(x).
In symbols, we use the known fact
and let u=g(x), giving us
an integral we can solve more easily:
Then, we substitute g(x) back in to get our desired result of kf(g(x))+C.
When we make a substitution in a definite integral in the form
we need to remember that the bounds are from x=a to x=b:
We then have two choices:
Plug x=a and x=b into u to find the bounds in terms of u.
Plug u(x) back in and use the bounds a→b.
Consider the integral
we can break the fraction into two parts:
Then, find A and B using a system of equations. Finally, integrate the two fractions.
To perform integration by parts, we use
Another common way of writing this is
Begin integration by parts by picking u and dv. Then, solve by differentiating u to get du and taking the anti-derivative of dv to get v.
Certain functions, like powers of x, will simplify after repeated differentiation.
Therefore, we may perform repeated integration by parts using such functions as our u knowing that they will eventually yield a simple ∫vdu.
Cyclical integration by parts are the result of a product of two functions that do not reduce via differentiation. Instead, they "cycle," or eventually return the original integral as the ∫vdu term.
In such cases, integrate by parts until you cycle back to the original integral I. Then, you may solve for I algebraically.