166 SCW, 11.4: Work for Lecture 20 Example 4

Integrate
x
5
x
+4
,x
Out[]=
1
40
1/5
2
2
2(5+
5
)
ArcTan
-1-
5
+2
3/5
2
x
10-2
5
-2
10-2
5
ArcTan
-1+
5
+2
3/5
2
x
2(5+
5
)
-4Log[2+
3/5
2
x]+(1+
5
)Log2+
(-1+
5
)x
2/5
2
+
1/5
2
2
x
-(-1+
5
)Log2-
(1+
5
)x
2/5
2
+
1/5
2
2
x

(%/.x->t)-(%/.x->n)
Out[]=
-
1
40
1/5
2
2
2(5+
5
)
ArcTan
-1-
5
+2
3/5
2
n
10-2
5
-2
10-2
5
ArcTan
-1+
5
+2
3/5
2
n
2(5+
5
)
-4Log[2+
3/5
2
n]+(1+
5
)Log2+
(-1+
5
)n
2/5
2
+
1/5
2
2
n
-(-1+
5
)Log2-
(1+
5
)n
2/5
2
+
1/5
2
2
n
+
1
40
1/5
2
2
2(5+
5
)
ArcTan
-1-
5
+2
3/5
2
t
10-2
5
-2
10-2
5
ArcTan
-1+
5
+2
3/5
2
t
2(5+
5
)
-4Log[2+
3/5
2
t]+(1+
5
)Log2+
(-1+
5
)t
2/5
2
+
1/5
2
2
t
-(-1+
5
)Log2-
(1+
5
)t
2/5
2
+
1/5
2
2
t

Limit[%,t->∞]
Out[]=
-
1
40
7/10
2
2
5-
5
π-2
5+
5
π+4
5+
5
ArcTan
-1-
5
+2
3/5
2
n
10-2
5
-4
5-
5
ArcTan
-1+
5
+2
3/5
2
n
2(5+
5
)
+
2
Log[4]-4
2
Log[2+
3/5
2
n]+
2
Log2+
(-1+
5
)n
2/5
2
+
1/5
2
2
n
+
10
Log2+
(-1+
5
)n
2/5
2
+
1/5
2
2
n
+
2
Log2-
(1+
5
)n
2/5
2
+
1/5
2
2
n
-
10
Log2-
(1+
5
)n
2/5
2
+
1/5
2
2
n

In[]:=
f[x_]:=
x
5
x
+4
f'[x]
Out[]=
-
5
5
x
2
(4+
5
x
)
+
1
4+
5
x
Simplify[%]
Out[]=
-
4(-1+
5
x
)
2
(4+
5
x
)
In[]:=
R[n_]:=-
1
40
7/10
2
2
5-
5
π-2
5+
5
π+4
5+
5
ArcTan
-1-
5
+2
3/5
2
n
10-2
5
-4
5-
5
ArcTan
-1+
5
+2
3/5
2
n
25+
5

+
2
Log[4]-4
2
Log[2+
3/5
2
n]+
2
Log2+
-1+
5
n
2/5
2
+
1/5
2
2
n
+
10
Log2+
-1+
5
n
2/5
2
+
1/5
2
2
n
+
2
Log2-
1+
5
n
2/5
2
+
1/5
2
2
n
-
10
Log2-
1+
5
n
2/5
2
+
1/5
2
2
n

R[10.]
Out[]=
0.000333328
Integrate
1
4
x
,{x,n,∞}
Out[]=
1
3
3
n
(1/(3*0.001))^(1/3)
Out[]=
6.93361
Sum
k
5
k
+4
,{k,1,7}
Out[]=
5345879867737369
19483637836543335
N[%]
Out[]=
0.274378
Sum
k
5
k
+4
,{k,1,∞}
Out[]=
-
1
5
RootSum5+5#1+10
2
#1
+10
3
#1
+5
4
#1
+
5
#1
&,
PolyGamma[0,-#1]
1+3#1+3
2
#1
+
3
#1
&
N[%]
Out[]=
0.275161+0.
?PolyGamma
Out[]=
Symbol
PolyGamma[z] gives the digamma function ψ(z). ​PolyGamma[n,z] gives the n
th
derivative of the digamma function
(n)
ψ
(z).