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25 Dec 2021Mathchan is launched into public

7 / 1 / 2 / ?

File: she's doing integrals ok.jpg ( 21.76 KB , 480x480 , 1655484028377.jpg )

Image
Primitive function of function
f(x)f(x)
over some interval
x[a,b]x\in[a, b]
is a function
F(x)F(x)
whose derivative is the function
f(x)f(x)
on that interval.

x[a,b],F(x)=f(x)\qquad \forall x\in[a, b],\quad F'(x) = f(x)


Antiderivative (aka indefinite integral) of a function
f(x)f(x)
is a family of its primitive functions which differ by a constant
CRC\in\mathbb{R}
:

f(x)dx=F(x)+C\qquad \int f(x)\mathrm{d}x = F(x) + C


In a nutshell, integration is the opposite of differentiation:

df(x)dxdx=F(x)\qquad \frac{\mathrm{d}\int f(x)\mathrm{d}x}{\mathrm{d}x} = F(x)


df(x)dx=f(x)dx\qquad \mathrm{d}\int f(x)\mathrm{d}x = f(x)\mathrm{d}x


dF(x)=F(x)+C\qquad \int\mathrm{d}F(x) = F(x) + C


Solving integrals in general is pretty hard, but there are a lot of established ways to do it. As OP I'll post some of the standard approaches, but this thread is about any kind of integration so feel free to post integrals and theirs solutions.

Most basic methods are:

  1. Using the table of integrals
  2. Using linearity property
  3. Using substitution
  4. Using partial integration
  5. Reducing quadratic to its cannonical form
  6. Partial decomposition
>>
Integration table

Exponents and Powers (special case for
xx
is obtained by setting
a=1,b=0a=1, b=0
)
(ax+b) dx=ax22+bx+C\int (ax + b)\ \mathrm{d}x = a\frac{x^2}{2} + bx + C
(ax+b)n dx=1a(ax+b)n+1n+1+C (n1)\int (ax + b)^n\ \mathrm{d}x = \frac{1}{a}\cdot \frac{(ax+b)^{n + 1}}{n + 1} + C\ (n\neq -1)
dxax+b=1alnax+b+C\int\frac{\mathrm{d}x}{ax + b} = \frac{1}{a}\ln|ax + b| + C
eax+bdx=ebaeax+b+C\int e^{ax + b}\mathrm{d}x = \frac{e^b}{a}\cdot e^{ax+b} + C

Trigonometric:

sin(ωx)dx=1ωcos(x)+C\int\sin(\omega x)\mathrm{d}x = -\frac{1}{\omega}\cos(x) + C
cos(ωx)dx=1ωsin(x)+C\int\cos(\omega x)\mathrm{d}x = \frac{1}{\omega}\sin(x) + C
tan(ωx)dx=1ωln1cos(x)+C\int\tan(\omega x)\mathrm{d}x = \frac{1}{\omega}\ln\left|\frac{1}{\cos(x)}\right| + C
dxsin2(ωx)=1ωcot(ωx)+C\int\frac{\mathrm{d}x}{\sin^2(\omega x)} = -\frac{1}{\omega}\cot(\omega x) + C
dxcos2(ωx)=1ωtan(ωx)+C\int\frac{\mathrm{d}x}{\cos^2(\omega x)} = \frac{1}{\omega}\tan(\omega x) + C

Hyperbolic:

sinh(x)dx=cosh(x)+C\int\sinh(x)\mathrm{d}x = \cosh(x) + C
cosh(x)dx=sinh(x)+C\int\cosh(x)\mathrm{d}x = \sinh(x) + C
tanh(x)dx=lncosh(x)+C\int\tanh(x)\mathrm{d}x =\ln\cosh(x) + C
dxsinh2(x)=coth(x)+C\int\frac{\mathrm{d}x}{\sinh^2(x)} = -\coth(x) + C
dxcosh2(x)=tanh(x)+C\int\frac{\mathrm{d}x}{\cosh^2(x)} = \tanh(x) + C


Inverse quadratic:

dxx2+a2=1aarctan(xa)+C\int\frac{\mathrm{d}x}{x^2 + a^2} = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C
dxx2a2=12alnxax+a+C\int\frac{\mathrm{d}x}{x^2 - a^2}=\frac{1}{2a}\ln\left|\frac{x - a}{x + a}\right| +C
dxa2x2=12alna+xax+C\int\frac{\mathrm{d}x}{a^2 - x^2} = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C
dxx2+a2=lnx+x2+a2+C\int\frac{\mathrm{d}x}{\sqrt{x^2 + a^2}} = \ln\left|x + \sqrt{x^2 + a^2}\right| + C
dxx2a2=lnx+x2a2+C\int\frac{\mathrm{d}x}{\sqrt{x^2 - a^2}} = \ln\left|x + \sqrt{x^2 - a^2}\right| + C
dxa2x2=arcsin(xa)+C\int\frac{\mathrm{d}x}{\sqrt{a^2 - x^2}} = \arcsin\left(\frac{x}{a}\right) + C
>>
Linearity property

Integration is linear. This means the integral can "go through" a sum or a multiplication with a constant.

(αf(x)+βg(x))dx=αf(x)dx+βg(x)dx\qquad\int\left(\alpha f(x) + \beta g(x)\right)\mathrm{d}x = \alpha\int f(x)\mathrm{d}x + \beta\int g(x)\mathrm{d}x


(k=1nαkfk(x))dx=k=1nαkfk(x)dx\qquad\int\left(\sum_{k=1}^n \alpha_k f_k(x)\right)\mathrm{d}x = \sum_{k=1}^n\alpha_k\int f_k(x)\mathrm{d}x


Using this property, it's possible to split an integral into manysmaller integrals which can be solved independently using other methods (e.g. sing table)

Example:

(5x+7sin(2x)+x8)dx=5dxx+7sin(2x)dx+x8dx=5lnx72cos(2x)+x99+C\qquad\int\left(\frac{5}{x} + 7\sin(2x) + x^8\right)\mathrm{d}x = 5\int\frac{\mathrm{d}x}{x} + 7\int\sin(2x)\mathrm{d}x + \int x^8\mathrm{d}x = 5\ln|x| -\frac{7}{2}\cos(2x) + \frac{x^9}{9} +C
>>
Substitution

Let's say we want to integrate
f(x)f(x)
in order obtain its primitve function
F(x)F(x)
over some interval
[a,b][a,b]
:

f(x)dx=F(x)+C,x[a,b]\qquad \int f(x)\mathrm{d}x = F(x) + C,\quad x\in[a, b]


Now let's suppose we can identify a surjective function
g(x)g(x)
that is continuous on
[a,b][a,b]
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\qquad \int f(g(x))\cdot g'(x)\mathrm{d}x = \int f(g(x))\ \mathrm{d}g(x) = F(g(x)) + C


In a nushell, we've substituted
xx
with
g(x)g(x)
and solved the integral in the usual way.
The tricky part is identifying
g(x)g'(x)
. 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
d\mathrm{d}
in the OP is
df(x)=f(x)dx\mathrm{d}f(x) = f'(x)\mathrm{d}x
.
That means we could write
g(x)dxg'(x)\mathrm{d}x
as
dg(x)\mathrm{d}g(x)
:

f(g(x))g(x)dx=f(g(x)) dg(x)\qquad \int f(g(x))\cdot g'(x)\mathrm{d}x = \int f(g(x))\ \mathrm{d}g(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)=sin2(x)2+C\qquad\int\sin(x)\cos(x)\mathrm{d}x = \int\sin(x)\ \mathrm{d}\sin(x) = \frac{\sin^2(x)}{2} + C


Basically, treating
sin(x)\sin(x)
like
xx
and using
xdx=x22+C\int x\mathrm{d}x = \frac{x^2}{2} + C
.

The cringe and bluepilled way of solving this, of course, is recognizing
sin(2x)=2sin(x)cos(x)\sin(2x) = 2\sin(x)\cos(x)


sin(x)cos(x)dx=12sin(2x)dx=cos(2x)4+C=2sin2(x)14+C=sin2(x)2+(14+C)constant=sin2(x)2+C\qquad\int\sin(x)\cos(x)\mathrm{d}x = \frac{1}{2}\int\sin(2x)\mathrm{d}x = \frac{-\cos(2x)}{4} + C = \frac{2\sin^2(x) - 1}{4} + C = \frac{\sin^2(x)}{2} + \underbrace{\left(-\frac{1}{4} + C\right)}_{\text{constant}} = \frac{\sin^2(x)}{2} + C
>>
Partial integration

This basically allows you to swap what you integrate and what you integrate by.

u dv=uvv du+C\qquad\int u\ \mathrm{d}v = uv - \int v\ \mathrm{d}u + C



Usually, you use it to solve an integral with polynomial multiplied by one of the non-polynomial functions.

Pn(x)eaxsin(ax)cos(ax)ln(ax)arcsin(ax)arccos(ax)dx\qquad \int P_n(x)\cdot \begin{matrix}e^{ax}\\\sin(ax)\\\cos(ax)\\\ln(ax)\\\arcsin(ax)\\\arccos(ax)\end{matrix}\mathrm{d}x


You usually choose
uu
to be the non-polynomial because calculating the derivative of it is probably going to be easier than integrating it.
On the other hand, polynomials are easy to both derive and integrate.

Example

You integrate
x2arccos(x)dx\int x^2\arccos(x)\mathrm{d}x
by differentiating
arccos(x)\arccos(x)
and by integrating
x2x^2
:

arccos(x)x2dx={u=arccos(x)du=dx1x2dv=x2dxv=x33}=x33arccos(x)+13x31x2dx+C=x33arccos(x)13(1x2+(1x2)33)+C\begin{aligned} \qquad \int \arccos(x)\cdot x^2 \mathrm{d}x &= \begin{Bmatrix} u = \arccos(x) & \mathrm{d}u = -\frac{\mathrm{d}x}{\sqrt{1- x^2}}\\\mathrm{d}v = x^2\mathrm{d}x & v = \frac{x^3}{3}\end{Bmatrix} = \frac{x^3}{3}\arccos(x)+ \frac{1}{3}\int\frac{x^3}{\sqrt{1-x^2}}\mathrm{d}x +C \\&= \frac{x^3}{3}\arccos(x) - \frac{1}{3}\left(\sqrt{1-x^2} + \frac{\sqrt{(1-x^2)^3}}{3}\right) + C \end{aligned}


As for how you solve
x31x2dx\int\frac{x^3}{\sqrt{1-x^2}}\mathrm{d}x
you do it by substituting
t=1x2,x2=1t2,2dx=2tdtt = \sqrt{1-x^2},\quad x^2 = 1- t^2,\quad \cancel{2}\mathrm{d}x = -\cancel{2}t\mathrm{d}t
:

x31x2dx=1t2t(t)dt=tt33=1x2(1x2)33\qquad \int\frac{x^3}{\sqrt{1-x^2}}\mathrm{d}x = \int\frac{1-t^2}{t}(-t)\mathrm{d}t = t - \frac{t^3}{3} = \boxed{\sqrt{1-x^2} - \frac{\sqrt{(1-x^2)^3}}{3}}
>>
Quadratic trinomial

How do you solve
dxax2+bx+c\int\frac{\mathrm{d}x}{ax^2 + bx + c}
and
dxax2+bx+c\int\frac{\mathrm{d}x}{\sqrt{ax^2 + bx + c}}
? You write
ax2+bx+cax^2 + bx + c
in the following way (hint: completing the square by adding and subtracting
b24)\frac{b^2}{4})
:

ax2+bx+c=a(x2+bax+ca)=a((x2+2b2ax+b24a2)+(b24a2+c))=a((x+b2a)2+(cb2a)2)=a(t2+k2),t=x+b2a,k=cb2a\begin{aligned} \qquad ax^2 + bx + c &= a\left(x^2 + \frac{b}{a}x + \frac{c}{a}\right) = a\left(\left(x^2 + 2\frac{b}{2a}x + \frac{b^2}{4a^2}\right) +\left(- \frac{b^2}{4a^2} + c \right)\right)\\&= a\left(\left(x + \frac{b}{2a}\right)^2 +\left(\sqrt{c - \frac{b}{2a}}\right)^2\right)\\&=a(t^2 + k^2),\quad t= x+ \frac{b}{2a},\quad k=\sqrt{c-\frac{b}{2a}} \end{aligned}


Now the integral reduces to either
1adxt2+k2=1akarctantk+C\boxed{\frac{1}{a}\int\frac{\mathrm{d}x}{t^2 + k^2} = \frac{1}{ak}\arctan{\frac{t}{k}} + C}
or
1adxt2+k2=1alnt2+t2+k2+C\boxed{\frac{1}{a}\int\frac{\mathrm{d}x}{\sqrt{t^2 + k^2}} = \frac{1}{a}\ln\left|t^2 + \sqrt{t^2 + k^2}\right| + C}
>>
Partial fraction decomposition

How do you solve
P(x)Q(x)dx\int\frac{P(x)}{Q(x)}\mathrm{d}x
where
 degP(x)<degQ(x)\ \deg{P(x)} < \deg{Q(x)}
?


First, as a consequence of the fundamental theorem of algebra, any real polynomial can be factored into linear and quadtratic terms.
We will do that with
Q(x)Q(x)
:

Q(x)=(xa1)A1(xa2)A2(xam)Am(x2+b1x+c1)B1(x2+b2x+c2)B2(x2+bnx+cn)Bn\qquad Q(x) = (x-a_1)^{A_1}(x-a_2)^{A_2}\dots(x - a_m)^{A_m}(x^2 + b_1x + c_1)^{B_1}(x^2 + b_2 x + c_2)^{B_2}\dots(x^2 + b_n x + c_n)^{B_n}


Now employ the partial fraction decomposition:

P(x)Q(x)=i=1mj=1Aiaij(xai)j+i=1nj=1Bibijx+cij(x2+bix+)j\qquad\frac{P(x)}{Q(x)} = \sum_{i=1}^m\sum_{j=1}^{A_i}\frac{a_{ij}}{(x-a_i)^j} + \sum_{i=1}^n\sum_{j=1}^{B_i}\frac{b_{ij}x + c_{ij}}{(x^2 + b_ix +)^j}


Then just use the linearity property of the integral.

For example:

x2+1x53x4+x3+7x26x8dx=x2+1(x2)(x+1)2(x23x+4)dx=a11x2dx+a21x+1dx+a22(x+1)2dx+b11x+c11x23x+4dx\qquad \int\frac{x^2 + 1}{x^5 - 3x^4 + x^3 + 7x^2 - 6x - 8}\mathrm{d}x = \int\frac{x^2 + 1}{(x-2)(x +1)^2(x^2-3x + 4)}\mathrm{d}x = \int \frac{a_{11}}{x-2}\mathrm{d}x + \int\frac{a_{21}}{x+1}\mathrm{d}x + \int\frac{a_{22}}{(x+1)^2}\mathrm{d}x + \int\frac{b_{11} x + c_{11}}{x^2 - 3x + 4} \mathrm{d}x



The constants
aij,bij,cijRa_{ij},b_{ij}, c_{ij}\in\mathbb{R}
have to be found e.g. using the Heaviside cover up method.
>>
cool nad thanks