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Find the integral $\int\frac{2x^3-4x^2-15x+15}{x^2-2x+8}dx$

Step-by-step Solution

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Final Answer

$x^2+31\ln\left(\frac{\sqrt{7}}{\sqrt{7+\left(x-1\right)^2}}\right)-\frac{16\sqrt{7}}{7}\arctan\left(\frac{\sqrt{7}}{7}\left(x-1\right)\right)+C_0$
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Step-by-step Solution

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Divide $2x^3-4x^2-15x+15$ by $x^2-2x+8$

$\begin{array}{l}\phantom{\phantom{;}x^{2}-2x\phantom{;}+8;}{\phantom{;}2x\phantom{;}\phantom{-;x^n}}\\\phantom{;}x^{2}-2x\phantom{;}+8\overline{\smash{)}\phantom{;}2x^{3}-4x^{2}-15x\phantom{;}+15\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+8;}\underline{-2x^{3}+4x^{2}-16x\phantom{;}\phantom{-;x^n}}\\\phantom{-2x^{3}+4x^{2}-16x\phantom{;};}-31x\phantom{;}+15\phantom{;}\phantom{;}\\\end{array}$

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$\begin{array}{l}\phantom{\phantom{;}x^{2}-2x\phantom{;}+8;}{\phantom{;}2x\phantom{;}\phantom{-;x^n}}\\\phantom{;}x^{2}-2x\phantom{;}+8\overline{\smash{)}\phantom{;}2x^{3}-4x^{2}-15x\phantom{;}+15\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+8;}\underline{-2x^{3}+4x^{2}-16x\phantom{;}\phantom{-;x^n}}\\\phantom{-2x^{3}+4x^{2}-16x\phantom{;};}-31x\phantom{;}+15\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve problems step by step online. Find the integral int((2x^3-4x^2-15x+15)/(x^2-2x+8))dx. Divide 2x^3-4x^2-15x+15 by x^2-2x+8. Resulting polynomial. Expand the integral \int\left(2x+\frac{-31x+15}{x^2-2x+8}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int2xdx results in: x^2.

Final Answer

$x^2+31\ln\left(\frac{\sqrt{7}}{\sqrt{7+\left(x-1\right)^2}}\right)-\frac{16\sqrt{7}}{7}\arctan\left(\frac{\sqrt{7}}{7}\left(x-1\right)\right)+C_0$

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Function Plot

Plotting: $x^2+31\ln\left(\frac{\sqrt{7}}{\sqrt{7+\left(x-1\right)^2}}\right)-\frac{16\sqrt{7}}{7}\arctan\left(\frac{\sqrt{7}}{7}\left(x-1\right)\right)+C_0$

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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

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