Find the integral $\int\frac{3-x}{x^3+4x^2+3x}dx$

Step-by-step Solution

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Final answer to the problem

$\ln\left|x\right|+\ln\left|x+3\right|-2\ln\left|x+1\right|+C_0$
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Step-by-step Solution

How should I solve this problem?

  • Integrate by parts
  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
  • FOIL Method
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We can factor the polynomial $x^3+4x^2+3x$ using the rational root theorem, which guarantees that for a polynomial of the form $a_nx^n+a_{n-1}x^{n-1}+\dots+a_0$ there is a rational root of the form $\pm\frac{p}{q}$, where $p$ belongs to the divisors of the constant term $a_0$, and $q$ belongs to the divisors of the leading coefficient $a_n$. List all divisors $p$ of the constant term $a_0$, which equals $0$

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Learn how to solve problems step by step online. Find the integral int((3-x)/(x^3+4x^23x))dx. We can factor the polynomial x^3+4x^2+3x using the rational root theorem, which guarantees that for a polynomial of the form a_nx^n+a_{n-1}x^{n-1}+\dots+a_0 there is a rational root of the form \pm\frac{p}{q}, where p belongs to the divisors of the constant term a_0, and q belongs to the divisors of the leading coefficient a_n. List all divisors p of the constant term a_0, which equals 0. Next, list all divisors of the leading coefficient a_n, which equals 1. The possible roots \pm\frac{p}{q} of the polynomial x^3+4x^2+3x will then be. We can factor the polynomial x^3+4x^2+3x using synthetic division (Ruffini's rule). We found that -1 is a root of the polynomial.

Final answer to the problem

$\ln\left|x\right|+\ln\left|x+3\right|-2\ln\left|x+1\right|+C_0$

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Function Plot

Plotting: $\ln\left(x\right)+\ln\left(x+3\right)-2\ln\left(x+1\right)+C_0$

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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
|◻|
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

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