These are my complete notes for U-Substitution in Integral Calculus.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@berkeley.edu.
Summary of U-Substitution (Integral Calculus)
?. U-Substitution.
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Rule .
Whenever you have an indefinite integral with a variable of integration different from the variable in the expression, you just have to plug in whatever value u or x is equal to into the expression to make it agree with the variable of integration. For example, where u = x²:
$$\int (u^3 + 1)dx = \int (x^6 + 1) dx$$
$$\int (u^3 + 1)dx = \int (x^6 + 1) dx$$
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Rule .
Sometimes you don't want to plug in the value for u and you actually want to replace the variable of integration (the dx or du). The move here is to take the value of u, u=2x for example, and take the derivative of both sides: du/dx = 2. Then, multiply by dx or du, where du = 2dx, and then isolate dx. dx = du/2. Finally, plug that in for the dx in the thing.
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Rule .
For the substitution -u business, you'll generally want u to be the exponent or wahtever makes the equation 'weird' (as in, the aspect that complicates the equation the most to not fit into one of the basic "PROPERTIES OF INDEFINITE INTEGRALS" molds). From there, take the derivative and do everything necessary to isolate dx so that the front expression can all be one variable. Then, take the antidervative as normal (since it should now have been converted into the basic molds showcased in the properties), and when you have completed it, substitute back in the value you set for u.
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Rule .
As a final addendum to u-sub's, take into consideration how they are meant to remove as much as possible from the expression in the name of u, so how du and dx are isolated can be messed around with to be whatever. If you see anything outside of the u (which is just anything that looks chain-ruled), you can warp the middle du/dx equation to reflect what you went to take out: ⅙du = x²dx.
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Rule .
When the x-variable in the integral are to the same degree, you must solve for x. When creating u, instead of taking the derivative right away, isolate x. For example, in a definite integral like
$$\int x \sqrt{3x+2} dx$$
, where u = 3x+2, you can establish that u-2=3x.
From there, (u-2) / 3 = x. You now have what u is equal to, so you can carry on with the rest of the substitution as normal: (du / 3) = dx. By distributing u in the expression, you will create multiple expressions in the solutions.
From there, (u-2) / 3 = x. You now have what u is equal to, so you can carry on with the rest of the substitution as normal: (du / 3) = dx. By distributing u in the expression, you will create multiple expressions in the solutions.
