# week4

## 1403 days ago by wldnd1217

var('x,y,a,b') f(x)=x^2 F(x)=integral(x^2, x) F(x)
 1/3*x^3 1/3*x^3
f(x)=(arctan(x/a)-arctan(x/b)) df(x)=diff(f(x),x) df(x)
 -1/((x^2/b^2 + 1)*b) + 1/((x^2/a^2 + 1)*a) -1/((x^2/b^2 + 1)*b) + 1/((x^2/a^2 + 1)*a)
f(x)=(tanh(x)) integral(f(x),x)
 log(cosh(x)) log(cosh(x))
f(x)=((x-a)*sqrt(2*a*x-x^2)+a^2*arcsin((x-a)/a)) diff(f(x),x)
 -(a - x)^2/sqrt(2*a*x - x^2) + a/sqrt(-(a - x)^2/a^2 + 1) + sqrt(2*a*x - x^2) -(a - x)^2/sqrt(2*a*x - x^2) + a/sqrt(-(a - x)^2/a^2 + 1) + sqrt(2*a*x - x^2)
var('x,y') f(x,y)=x^(3/2)-y^4-x^2*sin(y)-1 dxf(x,y)=diff(f(x,y),x) dyf(x,y)=diff(f(x,y),y) dxyf(x,y)=dxf(x,y)/dyf(x,y) dxyf(1,0)
 -3/2 -3/2
diff(sinh(tan(x))
 Traceback (click to the left of this block for traceback) ... SyntaxError: invalid syntax Traceback (most recent call last): File "", line 1, in File "_sage_input_32.py", line 10, in exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("ZGlmZihzaW5oKHRhbih4KSk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py")) File "", line 1, in File "/tmp/tmpgRCrU0/___code___.py", line 3 ^ SyntaxError: invalid syntax
var('x') y = function('y',x) de = diff(y,x) == 3*x^2 desolve(de,[y,x])
 x^3 + c x^3 + c
var('x') y = function('y',x) de = diff(y,x) == sec(x)*tan(x) desolve(de,[y,x])
 c + 1/cos(x) c + 1/cos(x)
y=function('y',x) de=diff(y,x)==((x+3*x^2)/(y^2)) desolve(de,y,ics=[0,6])
 1/3*y(x)^3 == x^3 + 1/2*x^2 + 72 1/3*y(x)^3 == x^3 + 1/2*x^2 + 72
#1 var('x,y,a,b') f(x)=(arctan(x/a)-arctan(x/b)) diff(f(x),x)
 -1/((x^2/b^2 + 1)*b) + 1/((x^2/a^2 + 1)*a) -1/((x^2/b^2 + 1)*b) + 1/((x^2/a^2 + 1)*a)
#2 f(x)=(arccos(x))^arctan(x) diff(f(x),x)
 (log(arccos(x))/(x^2 + 1) - arctan(x)/(sqrt(-x^2 + 1)*arccos(x)))*arccos(x)^arctan(x) (log(arccos(x))/(x^2 + 1) - arctan(x)/(sqrt(-x^2 + 1)*arccos(x)))*arccos(x)^arctan(x)
#3 f(x)=x^arctan(x) diff(f(x),x)
 (log(x)/(x^2 + 1) + arctan(x)/x)*x^arctan(x) (log(x)/(x^2 + 1) + arctan(x)/x)*x^arctan(x)
#4 f(x)=x*arctan(x) diff(f(x),x)
 x/(x^2 + 1) + arctan(x) x/(x^2 + 1) + arctan(x)
#5 f(x)=(x*sqrt(a^2-x^2)+a^2*arcsin(x/a)) diff(f(x),x)
 -x^2/sqrt(a^2 - x^2) + a/sqrt(-x^2/a^2 + 1) + sqrt(a^2 - x^2) -x^2/sqrt(a^2 - x^2) + a/sqrt(-x^2/a^2 + 1) + sqrt(a^2 - x^2)
#6 f(x)=(x*arccsc(1/x)+sqrt(1-x^2)) diff(f(x),x)
 arccsc(1/x) arccsc(1/x)
#7 f(x)=(x/sqrt(a^2-x^2)-arcsin(x/a)) diff(f(x),x)
 x^2/(a^2 - x^2)^(3/2) + 1/sqrt(a^2 - x^2) - 1/(sqrt(-x^2/a^2 + 1)*a) x^2/(a^2 - x^2)^(3/2) + 1/sqrt(a^2 - x^2) - 1/(sqrt(-x^2/a^2 + 1)*a)
#8 f(x)=((x-a)*sqrt(2*a*x-x^2)+a^2*arcsin(x/a-1)) diff(f(x),x)
 -(a - x)^2/sqrt(2*a*x - x^2) + a/sqrt(-(x/a - 1)^2 + 1) + sqrt(2*a*x - x^2) -(a - x)^2/sqrt(2*a*x - x^2) + a/sqrt(-(x/a - 1)^2 + 1) + sqrt(2*a*x - x^2)
#9 f(x)=(sqrt(x^2-4)/x^2+arccsc(x/2)/2) diff(f(x),x)
 1/(sqrt(x^2 - 4)*x) - 1/(sqrt(-4/x^2 + 1)*x^2) - 2*sqrt(x^2 - 4)/x^3 1/(sqrt(x^2 - 4)*x) - 1/(sqrt(-4/x^2 + 1)*x^2) - 2*sqrt(x^2 - 4)/x^3
#10 f(x,y)=y^2*sin(y)+y-arctan(x) df(x,y)=diff(f(x,y),x)/diff(f(x,y),y) df(x,y)
 -1/((x^2 + 1)*(y^2*cos(y) + 2*y*sin(y) + 1)) -1/((x^2 + 1)*(y^2*cos(y) + 2*y*sin(y) + 1))
#11 f(x)=e^x*cosh(x) diff(f(x),x)
 e^x*sinh(x) + e^x*cosh(x) e^x*sinh(x) + e^x*cosh(x)
#12 f(x)=sech(x)*(1+ln(sech(x))) diff(f(x),x)
 -(log(sech(x)) + 1)*tanh(x)*sech(x) - tanh(x)*sech(x) -(log(sech(x)) + 1)*tanh(x)*sech(x) - tanh(x)*sech(x)
#13 f(x)=e^(cosh(3*x)) diff(f(x),x)
 3*e^(cosh(3*x))*sinh(3*x) 3*e^(cosh(3*x))*sinh(3*x)
#14 f(x)=2*arctanh(tan(x/2)) diff(f(x),x)
 -(tan(1/2*x)^2 + 1)/(tan(1/2*x)^2 - 1) -(tan(1/2*x)^2 + 1)/(tan(1/2*x)^2 - 1)
#15 f(x)=a*arcsech(x/a)-sqrt(a^2-x^2) diff(f(x),x)
 x/sqrt(a^2 - x^2) - a/((x/a + 1)*x*sqrt(-(x/a - 1)/(x/a + 1))) x/sqrt(a^2 - x^2) - a/((x/a + 1)*x*sqrt(-(x/a - 1)/(x/a + 1)))
#16 f(x)=arcsinh(tan(x)) diff(f(x),x)
 sqrt(tan(x)^2 + 1) sqrt(tan(x)^2 + 1)
#17 f(x)=(x*arctanh(x)+ln(sqrt(1-x^2))) diff(f(x),x)
 -x/(-x^2 + 1) - x/(x^2 - 1) + arctanh(x) -x/(-x^2 + 1) - x/(x^2 - 1) + arctanh(x)
#18 f(x)=arcsech(e^(-x)) diff(f(x),x)
 1/((e^(-x) + 1)*sqrt(-(e^(-x) - 1)/(e^(-x) + 1))) 1/((e^(-x) + 1)*sqrt(-(e^(-x) - 1)/(e^(-x) + 1)))
#19 f(x)=arccoth(sec(x)) diff(f(x),x)
 -tan(x)*sec(x)/(sec(x)^2 - 1) -tan(x)*sec(x)/(sec(x)^2 - 1)
#20 f(x)=1/(x^4-1) integral(f(x),x)
 1/4*log(x - 1) - 1/4*log(x + 1) - 1/2*arctan(x) 1/4*log(x - 1) - 1/4*log(x + 1) - 1/2*arctan(x)
#21 f(x)=1/(x^3*sqrt(x^2-16)) integral(f(x),x)
 1/32*sqrt(x^2 - 16)/x^2 - 1/128*arcsin(4/abs(x)) 1/32*sqrt(x^2 - 16)/x^2 - 1/128*arcsin(4/abs(x))
#22 f(x)=(x^2+3*x+2)/sqrt(5-4*x-x^2) integral(f(x),x)
 -1/2*sqrt(-x^2 - 4*x + 5)*x - 9/2*arcsin(-1/3*x - 2/3) -1/2*sqrt(-x^2 - 4*x + 5)*x - 9/2*arcsin(-1/3*x - 2/3)
#23 f(x)=1/(sqrt(x)*(1-x^(1/3))) integral(f(x),x)
 -6*x^(1/6) - 3*log(x^(1/6) - 1) + 3*log(x^(1/6) + 1) -6*x^(1/6) - 3*log(x^(1/6) - 1) + 3*log(x^(1/6) + 1)
#24 f(x)=arctan(e^x)/e^x integral(f(x),x)
 -e^(-x)*arctan(e^x) - 1/2*log(e^(-2*x) + 1) -e^(-x)*arctan(e^x) - 1/2*log(e^(-2*x) + 1)
#25 f(x)=e^(2*x)*cos(3*x) integral(f(x),x)
 1/13*(3*sin(3*x) + 2*cos(3*x))*e^(2*x) 1/13*(3*sin(3*x) + 2*cos(3*x))*e^(2*x)