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Find the integral $\int\frac{6x^4+23x^2+4x+15}{x^2+3}dx$

Step-by-step Solution

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Final Answer

$2x^{3}+5x-4\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$
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Step-by-step Solution

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Divide $6x^4+23x^2+4x+15$ by $x^2+3$

$\begin{array}{l}\phantom{\phantom{;}x^{2}+3;}{\phantom{;}6x^{2}\phantom{-;x^n}+5\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3\overline{\smash{)}\phantom{;}6x^{4}\phantom{-;x^n}+23x^{2}+4x\phantom{;}+15\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3;}\underline{-6x^{4}\phantom{-;x^n}-18x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-6x^{4}-18x^{2};}\phantom{;}5x^{2}+4x\phantom{;}+15\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3-;x^n;}\underline{-5x^{2}\phantom{-;x^n}-15\phantom{;}\phantom{;}}\\\phantom{;-5x^{2}-15\phantom{;}\phantom{;}-;x^n;}\phantom{;}4x\phantom{;}\phantom{-;x^n}\\\end{array}$

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

$\begin{array}{l}\phantom{\phantom{;}x^{2}+3;}{\phantom{;}6x^{2}\phantom{-;x^n}+5\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3\overline{\smash{)}\phantom{;}6x^{4}\phantom{-;x^n}+23x^{2}+4x\phantom{;}+15\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3;}\underline{-6x^{4}\phantom{-;x^n}-18x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-6x^{4}-18x^{2};}\phantom{;}5x^{2}+4x\phantom{;}+15\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3-;x^n;}\underline{-5x^{2}\phantom{-;x^n}-15\phantom{;}\phantom{;}}\\\phantom{;-5x^{2}-15\phantom{;}\phantom{;}-;x^n;}\phantom{;}4x\phantom{;}\phantom{-;x^n}\\\end{array}$

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int((6x^4+23x^24x+15)/(x^2+3))dx. Divide 6x^4+23x^2+4x+15 by x^2+3. Resulting polynomial. Simplify the expression inside the integral. The integral \int6x^{2}dx results in: 2x^{3}.

Final Answer

$2x^{3}+5x-4\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$

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Function Plot

Plotting: $2x^{3}+5x-4\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$

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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: Integrals of Rational Functions

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

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