Use formula

polynomial 多锹式

Basically, there are two types of indefinite integral.

One is polynomial, which composed by ++ and āˆ’-

Another is composite function, which composed by Ɨ\times and Ć·\div

For the first type of integral, we use basic formula to each part of it directly:

∫3x2āˆ’2x+1xdx=∫(3xāˆ’2+1x)dx=32x2āˆ’2x+ln⁔∣x∣+C\begin{align*} &\int \frac{3x^2 - 2x + 1}{x} dx \\ \\ =& \int (3x - 2 + \frac{1}{x}) dx \\ \\ =& \frac{3}{2}x^2 - 2x + \ln{|x|} + C \end{align*}

For the second one, oh my god, it's very complicate.

Here's a basic template:

∫f(g(x))ā‹…g′(x)dx∫f(g(x))ā‹…dg(x)∫f(u)ā‹…du\int f(g(x)) \cdot g^\prime(x)dx \\ \\ \int f(g(x)) \cdot dg(x) \\ \\ \int f(u) \cdot du

Let's do some exercise:

∫xeāˆ’x2ā‹…dx=∫eāˆ’x2ā‹…xdx//compositeĀ functionĀ inĀ front=(āˆ’12)∫eāˆ’x2ā‹…(āˆ’2)xdx//findĀ aĀ wayĀ toĀ makeĀ xĀ =Ā (āˆ’x2)′=āˆ’2xĀ whileĀ keepingĀ equation’sĀ balance=(āˆ’12)∫eāˆ’x2ā‹…d(āˆ’x2)//weĀ knewĀ g′(x)dx=dg(x)=(āˆ’12)eāˆ’x2+C//seeĀ āˆ’x2Ā asĀ aĀ part,Ā thenĀ applyĀ basicĀ integralĀ formula\begin{align*} &\int xe^{-x^2} \cdot dx \\ \\ =& \int e^{-x^2} \cdot x dx &\text{//composite function in front} \\ \\ =& (-\frac{1}{2}) \int e^{-x^2} \cdot (-2)x dx &\text{//find a way to make x = } (-x^2)^\prime = -2x \text{ while keeping equation's balance} \\ \\ =& (-\frac{1}{2}) \int e^{-x^2} \cdot d(-x^2) &\text{//we knew }g^\prime(x)dx = dg(x) \\ \\ =& (-\frac{1}{2}) e^{-x^2} + C &\text{//see } -x^2 \text{ as a part, then apply basic integral formula} \end{align*}
∫cos⁔(1āˆ’3x)ā‹…dx=∫cos⁔(1āˆ’3x)ā‹…1dx//don’tĀ forgetĀ 1=(āˆ’13)∫cos⁔(1āˆ’3x)ā‹…(āˆ’3)1dx//findĀ aĀ wayĀ toĀ makeĀ 1Ā toĀ -3Ā whileĀ keepingĀ equation’sĀ balance=(āˆ’13)∫cos⁔(1āˆ’3x)ā‹…d(1āˆ’3x)//dydxdx=dy=(āˆ’13)sin⁔(1āˆ’3x)+C//seeĀ 1-3xĀ asĀ aĀ part,Ā thenĀ useĀ basicĀ formula\begin{align*} \\ \\ \\ &\int \cos(1-3x) \cdot dx \\ \\ =&\int \cos(1-3x) \cdot 1dx &\text{//don't forget 1} \\ \\ =& (-\frac{1}{3})\int \cos(1-3x) \cdot (-3)1dx &\text{//find a way to make 1 to -3 while keeping equation's balance} \\ \\ =& (-\frac{1}{3})\int \cos(1-3x) \cdot d(1-3x) &\text{//} \frac{dy}{dx} dx = dy \\ \\ =& (-\frac{1}{3}) \sin(1-3x) + C &\text{//see 1-3x as a part, then use basic formula} \\ \\ \\ \end{align*}

From these exercises, we know a criticle principle is: xx of dxdx is the original function, it can be just a xx, it also can be, for example, x2x^2.

∫(x)dx2=∫((x2)′⋅x)dx=∫(2xā‹…x)dx=∫(2x2)dx\begin{align*} \int (x) dx^2 = \int ((x^2)^\prime \cdot x) dx = \int (2x \cdot x) dx = \int (2x^2) dx \end{align*}
∫(1ā‹…x)dxwhere’sĀ 1Ā comeĀ from?Ā x′=1\begin{align*} \\ &\int (1 \cdot x) dx &\text{where's 1 come from? } x^\prime = 1 \end{align*}

Last updated

Was this helpful?