INTEGRATE..............
Answers
Step-by-step explanation:
One solution
1 + 2 = 12 + 9 = 21
2 + 3 = 23 + 13 = 36
3 + 4 = 34 + 9 = 43
4 + 5 = 45 + 13 = 58
a + b = ab + 9 / 13 alternatively
Another way
1 + 2 = 21 + 0 = 21
2 + 3 = 32 + 4 = 36
3 + 4 = 43 + 0 = 43
4 + 5 = 54 + 4 = 58
a + b = ba + 0 /4 alternatively
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Answer:
Strategy: Use Integration by Parts.
(integral)ln(x) dx
set
u = ln(x), dv = dx
then we find
du = (1/x) dx, v = x
substitute
(integral) ln(x) dx = (integral) u dv
and use integration by parts
= uv - (integral) v du
substitute u=ln(x), v=x, and du=(1/x)dx
= ln(x) x - (integral) x (1/x) dx
= ln(x) x - (integral) dx
= ln(x) x - x + C
= x ln(x) - x + C.
we will make use of the logarithimic integral which is a special function. Note that the logarithmic integral li(x) is defined as follows:
li(x)=∫x0dtln(t).
and it follows that the derivative of the logarithmic integral is 1ln(x) .
The first thing that we can do is take the derivative of ln(ln(x)) .
Using the chain rule, the derivative of ln(ln(x)) is 1xln(x).
Note that
ln(ln(x))=∫1xln(x)dx .
Now we will solve the integral ∫1xln(x)dx by using integration by parts. Let u=1x and dv=1ln(x)dx . Now, we get the following:
∫1xln(x)dx=(1x)(li(x))−∫li(x)∗−1x2dx
=li(x)x+∫li(x)x2dx
Thus, we have that
ln(ln(x))=li(x)x+∫li(x)x2dx
Now to get the integral we are looking for, notice that
∫ln(ln(x))dx=
∫li(x)xdx+∫(∫li(x)x2dx)dx
Now apply integration by parts on ∫(∫li(x)x2dx)dx to get the following:
Let u=∫li(x)x2dx and dv=dx . Now observe this
∫∫li(x)x2dx=
(∫li(x)x2dx)∗x−∫x∗(li(x)x2dx)
=(∫li(x)x2dx)∗x−∫(li(x)xdx)
Thus, we have the following:
∫ln(ln(x))=
∫li(x)xdx+(∫li(x)x2dx)∗x−∫li(x)xdx
=x∗(∫li(x)x2dx).
Note the following:
∫li(x)x2dx=ln(ln(x))−li(x)x
and thus, we have the following:
x∗(∫li(x)x2dx)=x∗(ln(ln(x))−li(x)x)
=xln(ln(x))−xli(x)x
=xln(ln(x))−li(x) .
Therefore, we have the following result:
∫ln(ln(x))dx=xln(ln(x))−li(x)+C .
Step-by-step explanation:
added Both........