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Find the integral $\int\frac{3x^2+6x+2}{x^2+3x+2}dx$

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Solution

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Final Answer

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

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Divide $3x^2+6x+2$ by $x^2+3x+2$

$\begin{array}{l}\phantom{\phantom{;}x^{2}+3x\phantom{;}+2;}{\phantom{;}3\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3x\phantom{;}+2\overline{\smash{)}\phantom{;}3x^{2}+6x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+2;}\underline{-3x^{2}-9x\phantom{;}-6\phantom{;}\phantom{;}}\\\phantom{-3x^{2}-9x\phantom{;}-6\phantom{;}\phantom{;};}-3x\phantom{;}-4\phantom{;}\phantom{;}\\\end{array}$

Learn how to solve integrals of rational functions problems step by step online.

$\begin{array}{l}\phantom{\phantom{;}x^{2}+3x\phantom{;}+2;}{\phantom{;}3\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3x\phantom{;}+2\overline{\smash{)}\phantom{;}3x^{2}+6x\phantom{;}+2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3x\phantom{;}+2;}\underline{-3x^{2}-9x\phantom{;}-6\phantom{;}\phantom{;}}\\\phantom{-3x^{2}-9x\phantom{;}-6\phantom{;}\phantom{;};}-3x\phantom{;}-4\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int((3x^2+6x+2)/(x^2+3x+2))dx. Divide 3x^2+6x+2 by x^2+3x+2. Resulting polynomial. Expand the integral \int\left(3+\frac{-3x-4}{x^2+3x+2}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int3dx results in: 3x.

Final Answer

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

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

Solve integral of ((3x^2+6x)/(x^2+3x))dx using partial fractionsSolve integral of ((3x^2+6x)/(x^2+3x))dx using basic integralsSolve integral of ((3x^2+6x)/(x^2+3x))dx using u-substitutionSolve integral of ((3x^2+6x)/(x^2+3x))dx using integration by parts

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

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

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
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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Main Topic: Integrals of Rational Functions

Integrals of rational functions of the form R(x) = P(x)/Q(x).

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