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# WORKSHEET#19
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# Integration in Maple
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> f:=1/(x^2+1);
1
f := ------
2
x + 1
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> int(f,x);
arctan(x)
--------------------------------------------------------------------------------
> diff(",x);
1
------
2
x + 1
--------------------------------------------------------------------------------
> Int(f,x);
/
| 1
| ------ dx
| 2
/ x + 1
--------------------------------------------------------------------------------
> Int(f,x)=int(f,x);
/
| 1
| ------ dx = arctan(x)
| 2
/ x + 1
--------------------------------------------------------------------------------
> Int(f,x)=int(f,x)+c;
/
| 1
| ------ dx = arctan(x) + c
| 2
/ x + 1
--------------------------------------------------------------------------------
> f:=1/sqrt(x^2-1);
1
f := -----------
2 1/2
(x - 1)
--------------------------------------------------------------------------------
> int(f,x);
2 1/2
ln(x + (x - 1) )
--------------------------------------------------------------------------------
> diff(",x);
x
1 + -----------
2 1/2
(x - 1)
---------------
2 1/2
x + (x - 1)
--------------------------------------------------------------------------------
> simplify(");
1
-----------
2 1/2
(x - 1)
--------------------------------------------------------------------------------
> f:=sqrt(1-x^2);
2 1/2
f := (1 - x )
--------------------------------------------------------------------------------
> int(f,x);
2 1/2
1/2 x (1 - x ) + 1/2 arcsin(x)
--------------------------------------------------------------------------------
> diff(",x);
2
2 1/2 x 1
1/2 (1 - x ) - 1/2 ----------- + -------------
2 1/2 2 1/2
(1 - x ) 2 (1 - x )
--------------------------------------------------------------------------------
> simplify(");
2 1/2
I (x - 1)
--------------------------------------------------------------------------------
> simplify("-f);
2 1/2 2 1/2
I (x - 1) - (1 - x )
--------------------------------------------------------------------------------
> simplify(",symbolic);
0
--------------------------------------------------------------------------------
> int(f,x=-1..1);
1/2 Pi
--------------------------------------------------------------------------------
> int(f,x=1..2);
1/2
I 3 + 1/2 arcsin(2) - 1/4 Pi
--------------------------------------------------------------------------------
> evalf((1/2)*arcsin(2));
.7853981635 - .6584789485 I
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> evalf("");
1.073571860 I
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> evalf(Pi/4);
.7853981635
--------------------------------------------------------------------------------
> subs(x=2,f);
1/2
(-3)
--------------------------------------------------------------------------------
> evalf(");
1.732050808 I
--------------------------------------------------------------------------------
> f:=1/sqrt(1-x^4);
1
f := -----------
4 1/2
(1 - x )
--------------------------------------------------------------------------------
> int(f,x);\
/
| 1
| ----------- dx
| 4 1/2
/ (1 - x )
--------------------------------------------------------------------------------
> evalf(int(f,x=0..1));
1.311028778
--------------------------------------------------------------------------------
> evalf(Int(f,x=0..1));
1.311028777
--------------------------------------------------------------------------------
> Digits:=20:
--------------------------------------------------------------------------------
> evalf(int(f,x=0..1));
1.3110287771460599053
--------------------------------------------------------------------------------
> evalf(Int(f,x=0..1));
1.3110287771460599052
--------------------------------------------------------------------------------
> Digits:=10:
--------------------------------------------------------------------------------
> int(f,x=0..1);
3/2 1/2
Pi 2
1/4 -----------
2
GAMMA(3/4)
--------------------------------------------------------------------------------
> evalf(");
1.311028778
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> Int(f,x=0..1);
1
/
| 1
| ----------- dx
| 4 1/2
/ (1 - x )
0
--------------------------------------------------------------------------------
> evalf(");
1.311028777
--------------------------------------------------------------------------------
> f:=(x^2+x+1)/(x^4-1);\
2
x + x + 1
f := ----------
4
x - 1
--------------------------------------------------------------------------------
> partialfrac(");
2
x + x + 1
partialfrac(----------)
4
x - 1
--------------------------------------------------------------------------------
> ?partialfrac
Try one of the following topics:
{parse, parfrac, parametric, parity, parallel, parampass, parameter, partition}
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> parfrac(f);
2
x + x + 1
parfrac(----------)
4
x - 1
--------------------------------------------------------------------------------
> ?parfrac
--------------------------------------------------------------------------------
> convert(f,parfrac,x);
3 1 x
--------- - --------- - 1/2 ------
4 (x - 1) 4 (x + 1) 2
x + 1
--------------------------------------------------------------------------------
> int(",x);
2
3/4 ln(x - 1) - 1/4 ln(x + 1) - 1/4 ln(x + 1)
--------------------------------------------------------------------------------
> diff(",x);
3 1 x
--------- - --------- - 1/2 ------
4 (x - 1) 4 (x + 1) 2
x + 1
--------------------------------------------------------------------------------
> f:=1/(x^3-1);
1
f := ------
3
x - 1
--------------------------------------------------------------------------------
> g:=int(f,x);
2 1/2 1/2
g := 1/3 ln(x - 1) - 1/6 ln(x + x + 1) - 1/3 3 arctan(1/3 (2 x + 1) 3 )
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> plot(g,x=1..4);
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#
** Maple V Graphics **
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> limit(g,x,right);
Error, (in limit) invalid arguments
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> ?limit
--------------------------------------------------------------------------------
> limit(g,x=1,right);
- infinity
--------------------------------------------------------------------------------
> limit(g,x=1,left);
- infinity
--------------------------------------------------------------------------------
> g(-2);
2
1/3 ln(x - 1)(-2) - 1/6 ln(x + x + 1)(-2)
1/2 1/2
- 1/3 3 arctan(1/3 (2 x + 1) 3 )(-2)
--------------------------------------------------------------------------------
> subs(x=-2,g);
1/2 1/2
1/3 ln(-3) - 1/6 ln(3) - 1/3 3 arctan(- 3 )
--------------------------------------------------------------------------------
> evalf(");
.7877018364 + 1.047197551 I
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> ln(-3);
ln(-3)
--------------------------------------------------------------------------------
> evalf(");
1.098612289 + 3.141592654 I
--------------------------------------------------------------------------------
> subs(x=2.0,g);
1/2 1/2
1/3 ln(1.0) - 1/6 ln(7.00) - 1/3 3 arctan(1.666666667 3 )
--------------------------------------------------------------------------------
> evalf(");
-1.038687215
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> limit(g,x=infinity);
1/2
- 1/6 3 Pi
--------------------------------------------------------------------------------
> evalf(");
-.9068996827
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> int(f,x=0..2);
2
/
| 1
| ------ dx
| 3
/ x - 1
0
--------------------------------------------------------------------------------
> evalf(");
Error, (in evalf/int) unable to handle singularity
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> int(f,x=0..0.9);
1/2
- 1.520747795 + 1.047197551 I + 1/18 3 Pi - 1/3 I Pi
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> evalf(");
-1.218447901
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> evalf(Int(f,x=0..0.9));
-1.218447901
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> int(f,x=2..infinity);
1/2 1/2 1/2
- 1/6 3 Pi + 1/6 ln(7) + 1/3 3 arctan(5/3 3 )
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> evalf(");
.1317875321
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> g:=int(exp(-x^2),x);
1/2
g := 1/2 Pi erf(x)
--------------------------------------------------------------------------------
> diff(g,x);
2
exp(- x )
--------------------------------------------------------------------------------
> subs(x=0,g);
1/2
1/2 Pi erf(0)
--------------------------------------------------------------------------------
> erf(0);
0
--------------------------------------------------------------------------------
> ?erf
--------------------------------------------------------------------------------
> int(exp(-x^2),x=0..infinity);
1/2
1/2 Pi
> erf(infinity);
1
--------------------------------------------------------------------------------
> int((sin(x))^4,x);
3
- 1/4 sin(x) cos(x) - 3/8 cos(x) sin(x) + 3/8 x
--------------------------------------------------------------------------------
> diff(",x);
2 2 4 2 2
- 3/4 sin(x) cos(x) + 1/4 sin(x) + 3/8 sin(x) - 3/8 cos(x) + 3/8
--------------------------------------------------------------------------------
> simplify(");
2 4
- 2 cos(x) + 1 + cos(x)
--------------------------------------------------------------------------------
> simplify("-(sin(x))^4);
0
--------------------------------------------------------------------------------
> int(sin(x)/x,x);
Si(x)
> ?Si
--------------------------------------------------------------------------------
> Si(0);
0
--------------------------------------------------------------------------------
> Si(infinity);
1/2 Pi
--------------------------------------------------------------------------------
> Int(sin(x)/x,x=0..infinity)=Si(infinity);
infinity
/
| sin(x)
| ------ dx = 1/2 Pi
| x
/
0
--------------------------------------------------------------------------------
> f:='f':
--------------------------------------------------------------------------------
> L(f)=Int(exp(-s*t)*f(t),t=0..infinity);
infinity
/
|
L(f) = | exp(- s t) f(t) dt
|
/
0
--------------------------------------------------------------------------------
> int(exp(-s*t),t=0..infinity);
exp(- s t)
limit - ---------- + 1/s
t -> infinity- s
--------------------------------------------------------------------------------
> assume(s>0);
--------------------------------------------------------------------------------
> int(exp(-s*t),t=0..infinity);
1
----
s~
--------------------------------------------------------------------------------
> about(s);
Originally s, renamed s~:
is assumed to be: RealRange(Open(0),infinity)
--------------------------------------------------------------------------------
> Int(exp(-s*t)*t^(1/2),t=0..infinity)=int(exp(-s*t)*t^(1/2),t=0..infinity);
infinity
/ 1/2
| 1/2 Pi
| exp(- s~ t) t dt = 1/2 -----
| 3/2
/ s~
0
--------------------------------------------------------------------------------
> Int(exp(-s*t)*exp(t),t=0..infinity)=int(exp(-s*t)*exp(t),t=0..infinity);
infinity
/
| exp(t) exp(- s~ t) 1
| exp(- s~ t) exp(t) dt = limit - ------------------ + ------
| t -> infinity- s~ - 1 s~ - 1
/
0
--------------------------------------------------------------------------------
> additionally(s>1);
--------------------------------------------------------------------------------
> Int(exp(-s*t)*exp(t),t=0..infinity)=int(exp(-s*t)*exp(t),t=0..infinity);
infinity
/
| 1
| exp(- s~ t) exp(t) dt = ------
| s~ - 1
/
0
--------------------------------------------------------------------------------
> about(s);
Originally s, renamed s~:
is assumed to be: RealRange(Open(1),infinity)
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> s:='s':
--------------------------------------------------------------------------------
> ?laplace
--------------------------------------------------------------------------------
> laplace(1,t,s);
1/s
--------------------------------------------------------------------------------
> laplace(t^(1/2),t,s);laplace(t*exp(t),t,s);laplace(exp(a*t),t,s);\
laplace(sin(b*t),t,s);
1/2
Pi
1/2 -----
3/2
s
1
--------
2
(s - 1)
1
-----
s - a
b
-------
2 2
s + b
--------------------------------------------------------------------------------
> ?invlaplace
--------------------------------------------------------------------------------
> invlaplace(s/(s^2+b^2),s,t);
cos(b t)
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> invlaplace(1/(s-1)^3,s,t);invlaplace(1/sqrt(s^2+b^2),s,t);\
invlaplace(exp(-s^2),s,t);
2
1/2 t exp(t)
BesselJ(0, b t)
2
invlaplace(exp(- s ), s, t)
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