Integrate the function $f\ln\left(\frac{x^5}{3x^2+6x+2}\right)$

Step-by-step Solution

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Final answer to the problem

$-3fx+5xf\ln\left|x\right|+\frac{f}{\sqrt{3}}\ln\left|\sqrt{3}x+\sqrt{3}-1\right|+\frac{-f}{\sqrt{3}}\ln\left|\sqrt{3}x+\sqrt{3}+1\right|-f\ln\left|-\frac{1}{3}+\left(x+1\right)^2\right|-fx\ln\left|3x^2+6x+2\right|+C_0$
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Step-by-step Solution

How should I solve this problem?

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

Find the integral

$\int f\ln\left(\frac{x^5}{3x^2+6x+2}\right)dx$

Learn how to solve differential calculus problems step by step online.

$\int f\ln\left(\frac{x^5}{3x^2+6x+2}\right)dx$

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Unlock the first 3 steps of this solution

Learn how to solve differential calculus problems step by step online. Integrate the function fln((x^5)/(3x^2+6x+2)). Find the integral. The integral of a function times a constant (f) is equal to the constant times the integral of the function. The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. Expand the integral \int\left(\ln\left(x^5\right)-\ln\left(3x^2+6x+2\right)\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately.

Final answer to the problem

$-3fx+5xf\ln\left|x\right|+\frac{f}{\sqrt{3}}\ln\left|\sqrt{3}x+\sqrt{3}-1\right|+\frac{-f}{\sqrt{3}}\ln\left|\sqrt{3}x+\sqrt{3}+1\right|-f\ln\left|-\frac{1}{3}+\left(x+1\right)^2\right|-fx\ln\left|3x^2+6x+2\right|+C_0$

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Function Plot

Plotting: $-3fx+5xf\ln\left(x\right)+\frac{f}{\sqrt{3}}\ln\left(\sqrt{3}x+\sqrt{3}-1\right)+\frac{-f}{\sqrt{3}}\ln\left(\sqrt{3}x+\sqrt{3}+1\right)-f\ln\left(-\frac{1}{3}+\left(x+1\right)^2\right)-fx\ln\left(3x^2+6x+2\right)+C_0$

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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

How to improve your answer:

Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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