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Find the integral $\int\frac{2x^3+7x^2+2x+9}{2x+3}dx$

Step-by-step Solution

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Final Answer

$\frac{x^{3}}{3}+x^2-2x+\frac{15}{2}\ln\left(2x+3\right)+C_0$
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Step-by-step Solution

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Divide $2x^3+7x^2+2x+9$ by $2x+3$

$\begin{array}{l}\phantom{\phantom{;}2x\phantom{;}+3;}{\phantom{;}x^{2}+2x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{;}2x\phantom{;}+3\overline{\smash{)}\phantom{;}2x^{3}+7x^{2}+2x\phantom{;}+9\phantom{;}\phantom{;}}\\\phantom{\phantom{;}2x\phantom{;}+3;}\underline{-2x^{3}-3x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-2x^{3}-3x^{2};}\phantom{;}4x^{2}+2x\phantom{;}+9\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x\phantom{;}+3-;x^n;}\underline{-4x^{2}-6x\phantom{;}\phantom{-;x^n}}\\\phantom{;-4x^{2}-6x\phantom{;}-;x^n;}-4x\phantom{;}+9\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x\phantom{;}+3-;x^n-;x^n;}\underline{\phantom{;}4x\phantom{;}+6\phantom{;}\phantom{;}}\\\phantom{;;\phantom{;}4x\phantom{;}+6\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}15\phantom{;}\phantom{;}\\\end{array}$

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$\begin{array}{l}\phantom{\phantom{;}2x\phantom{;}+3;}{\phantom{;}x^{2}+2x\phantom{;}-2\phantom{;}\phantom{;}}\\\phantom{;}2x\phantom{;}+3\overline{\smash{)}\phantom{;}2x^{3}+7x^{2}+2x\phantom{;}+9\phantom{;}\phantom{;}}\\\phantom{\phantom{;}2x\phantom{;}+3;}\underline{-2x^{3}-3x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-2x^{3}-3x^{2};}\phantom{;}4x^{2}+2x\phantom{;}+9\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x\phantom{;}+3-;x^n;}\underline{-4x^{2}-6x\phantom{;}\phantom{-;x^n}}\\\phantom{;-4x^{2}-6x\phantom{;}-;x^n;}-4x\phantom{;}+9\phantom{;}\phantom{;}\\\phantom{\phantom{;}2x\phantom{;}+3-;x^n-;x^n;}\underline{\phantom{;}4x\phantom{;}+6\phantom{;}\phantom{;}}\\\phantom{;;\phantom{;}4x\phantom{;}+6\phantom{;}\phantom{;}-;x^n-;x^n;}\phantom{;}15\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve problems step by step online. Find the integral int((2x^3+7x^22x+9)/(2x+3))dx. Divide 2x^3+7x^2+2x+9 by 2x+3. Resulting polynomial. Expand the integral \int\left(x^{2}+2x-2+\frac{15}{2x+3}\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int x^{2}dx results in: \frac{x^{3}}{3}.

Final Answer

$\frac{x^{3}}{3}+x^2-2x+\frac{15}{2}\ln\left(2x+3\right)+C_0$

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Plotting: $\frac{x^{3}}{3}+x^2-2x+\frac{15}{2}\ln\left(2x+3\right)+C_0$

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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
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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