# Props && Thms of Intergral
###### tags: `Calculus`
### Linearity
$$
Assume\;that\;f\;and\;g\;are\;Riemann\;integrable\;on\;\lbrack a,b\rbrack\\Then\;\int_a^b(f(x)+g(x))dx=\int_a^bf(x)dx+\int_a^bg(x)dx\\and\;\;\int_a^bcf(x)dx=c\int_a^bf(x)dx\;,\forall c\in\mathbb{R}\\\Rightarrow\int_a^b(f(x)-g(x))dx=\int_a^bf(x)dx-\int_a^bg(x)dx
$$
##
### Comparison
$$
Assume\;that\;f\;and\;g\;are\;Riemann\;integrable\;on\;\lbrack a,b\rbrack\\f(x)\leq g(x)\;,\forall x\in\lbrack a,b\rbrack\\\Rightarrow\int_a^bf(x)dx\leq\int_a^bg(x)dx
$$
##
### Others-1
$$
f\;is\;Riemann\;integrable\;on\;\lbrack a,b\rbrack\\-\left|f(x)\right|\leq f(x)\leq\left|f(x)\right|\;,\forall x\in\lbrack a,b\rbrack\\\Rightarrow-\int_a^b\left|f(x)\right|dx\leq\int_a^bf(x)dx\leq\int_a^b\left|f(x)\right|dx\\\Rightarrow\left|\int_a^bf(x)dx\right|\leq\int_a^b\left|f(x)\right|dx
$$
##
### Others-2
$$
Assume\;that\;f\;is\;Riemann\;integrable\;on\;\lbrack a,b\rbrack,\lbrack b,c\rbrack\\1.\;Then\;f\;is\;Riemann\;integrable\;on\;\lbrack a',b'\rbrack\;,\;where\;a\leq a'\leq b'\leq b\\2.\;f\;is\;Riemann\;integrable\;on\;\lbrack a,c\rbrack\\(\int_a^cf(x)dx=\int_a^bf(x)dx+\int_b^cf(x)dx)
$$
##
### Technique
$$
Let\;f_+(x)=\left\{\begin{array}{l}f(x)\;,if\;f(x)\geq0\\0\;,if\;f(x)<0\end{array}=\frac12(f(x)+\left|f(x)\right|)\right.\\f_-(x)=\left\{\begin{array}{l}f(x)\;,if\;f(x)\leq0\\0\;,if\;f(x)<0\end{array}=\frac12(f(x)-\left|f(x)\right|)\right.\\f(x)=f_+(x)+f_-(x)\\\Rightarrow\int_a^bf(x)dx=\int_a^bf_+(x)dx+\int_a^bf_-(x)dx
$$
##
### Fundamental Theorem(IMPORTANT!!!)
### Part1
$$
Assume\;that\;f\;is\;Riemann\;integrable\;on\;\lbrack a,b\rbrack\\A(x)=\int_a^xf(t)dt\;,\;x\in\lbrack a,b\rbrack\\A'(x)=\frac d{dx}\int_a^xf(t)dt=f(x)
$$
#### <Proof>
$$
Assume\;x,\;x+h\;\in\lbrack a,b\rbrack\\A(x+h)-A(x)=\int_x^{x+h}f(t)dt\\\exists\;c\in\lbrack x,x+h\rbrack\;s.t.\;\;A(x+h)-A(x)=f(c)\cdot h\\\Rightarrow\;\frac{A(x+h)-A(x)}h=f(c)\\\Rightarrow\lim_{h\rightarrow0}\;\frac{A(x+h)-A(x)}h=\lim_{h\rightarrow0}f(c)\\\Rightarrow A'(x)=\lim_{c\rightarrow x}f(c)\\\Rightarrow A'(x)=f(x)
$$
### Part2
$$
If\;f\;is\;continuous\;on\;\lbrack a,b\rbrack\\then\;\frac d{dx}\int_a^xf(t)dt\;=\;f(x)\;\forall x\in(a,b)\\In\;other\;words,\;\int_a^xf(t)dt\;is\;an\;antiderivative/indefinite\;integral\;of\;f\\Cor.\\Let\;F(x)\;be\;any\;antiderivative\;of\;f\\then\;\int_a^xf(t)dt-F(x)=cons\tan t=\int_a^af(t)dt-F(a)=-F(a)\\\Rightarrow\int_a^xf(t)dt=F(x)-F(a)\\In\;particular,\\\int_a^bf(t)dt=F(b)-F(a)=F(x)\vert_{x=a}^b
$$
### Example
$$
\frac d{dx}\int_a^{\sin x}e^tdt\;=\;\frac d{dx}F(\sin x)=F'(\sin x)\cdot\cos x\\=e^{\sin x}\cdot\cos x
$$