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# Integrate the function $\frac{\ln\left(x\right)^5}{4x}$ from $1$ to $25$

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##  Final answer to the problem

$\frac{\ln\left(25\right)^{6}}{24}$
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##  Step-by-step Solution 

How should I solve this problem?

• Choose an option
• Integrate by partial fractions
• Integrate by substitution
• Integrate by parts
• Integrate using tabular integration
• Integrate by trigonometric substitution
• Weierstrass Substitution
• Integrate using trigonometric identities
• Integrate using basic integrals
• Product of Binomials with Common Term
Can't find a method? Tell us so we can add it.
1

Take the constant $\frac{1}{4}$ out of the integral

$\frac{1}{4}\int_{1}^{25}\frac{\ln\left(x\right)^5}{x}dx$

Learn how to solve definite integrals problems step by step online.

$\frac{1}{4}\int_{1}^{25}\frac{\ln\left(x\right)^5}{x}dx$

Learn how to solve definite integrals problems step by step online. Integrate the function (ln(x)^5)/(4x) from 1 to 25. Take the constant \frac{1}{4} out of the integral. We can solve the integral \int\frac{\ln\left(x\right)^5}{x}dx by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it u), which when substituted makes the integral easier. We see that \ln\left(x\right) it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part. Now, in order to rewrite dx in terms of du, we need to find the derivative of u. We need to calculate du, we can do that by deriving the equation above. Isolate dx in the previous equation.

##  Final answer to the problem

$\frac{\ln\left(25\right)^{6}}{24}$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

SnapXam A2

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5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

###  Main Topic: Definite Integrals

Given a function f(x) and the interval [a,b], the definite integral is equal to the area that is bounded by the graph of f(x), the x-axis and the vertical lines x=a and x=b