# Step-by-step Solution

Go!
1
2
3
4
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
|◻|
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

## Step-by-step explanation

Problem to solve:

$\int\frac{\ln\left(x+1\right)}{x}dx$

Learn how to solve calculus problems step by step online.

$\int\frac{\ln\left(x+1\right)}{x}dx$

Learn how to solve calculus problems step by step online. Calculate the integral of int(((ln(x+1)/x))dx. Use the Taylor series for rewrite the function \ln\left(x+1\right) as an approximation: \displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n, with a=0. Here we will use only the first four terms of the serie. Split the fraction \frac{x+\frac{-x^{2}}{2}+\frac{2x^{3}}{6}+\frac{-6x^{4}}{24}}{x} inside the integral, in two terms with common denominator x. Simplifying. The integral \int1dx results in: x.

$x-\frac{1}{4}x^2+\frac{1}{9}x^{3}-\frac{1}{16}x^{4}+C_0$

### Problem Analysis

$\int\frac{\ln\left(x+1\right)}{x}dx$

Calculus

~ 0.22 seconds