Here we look at the plot of the integrand
In[]:=
fun[x_]:=Cos[x]Log[
2
x
+4]
In[]:=
Plot[fun[x],{x,0,100}]
Out[]=
20
40
60
80
100
-10
-5
5
10
Here we look at the integral of the integrand over x∈[0,∞)
In[]:=
derived=Integrate[fun[x],{x,0,∞}]
Out[]=
∞
∫
0
Cos[x]Log[4+
2
x
]x
In[]:=
derived//N
Out[]=
-0.425168
In[]:=
actual=-
π
Exp[1]^2
Out[]=
-
π
2

In[]:=
actual//N
Out[]=
-0.425168
In[]:=
{derived,actual}
Out[]=

∞
∫
0
Cos[x]Log[4+
2
x
]x,-
π
2

