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Find the integral $\int\frac{4x^3-2x^2+x+16}{x^2-2x+4}dx$

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Solution

Final Answer

$2x^2+6x+3\ln\left(\frac{\sqrt{3}}{\sqrt{3+\left(x-1\right)^2}}\right)-\frac{11\sqrt{3}}{3}\arctan\left(0.577348\left(x-1\right)\right)+C_0$
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Step-by-step Solution

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Divide $4x^3-2x^2+x+16$ by $x^2-2x+4$

$\begin{array}{l}\phantom{\phantom{;}x^{2}-2x\phantom{;}+4;}{\phantom{;}4x\phantom{;}+6\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-2x\phantom{;}+4\overline{\smash{)}\phantom{;}4x^{3}-2x^{2}+x\phantom{;}+16\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+4;}\underline{-4x^{3}+8x^{2}-16x\phantom{;}\phantom{-;x^n}}\\\phantom{-4x^{3}+8x^{2}-16x\phantom{;};}\phantom{;}6x^{2}-15x\phantom{;}+16\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+4-;x^n;}\underline{-6x^{2}+12x\phantom{;}-24\phantom{;}\phantom{;}}\\\phantom{;-6x^{2}+12x\phantom{;}-24\phantom{;}\phantom{;}-;x^n;}-3x\phantom{;}-8\phantom{;}\phantom{;}\\\end{array}$

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$\begin{array}{l}\phantom{\phantom{;}x^{2}-2x\phantom{;}+4;}{\phantom{;}4x\phantom{;}+6\phantom{;}\phantom{;}}\\\phantom{;}x^{2}-2x\phantom{;}+4\overline{\smash{)}\phantom{;}4x^{3}-2x^{2}+x\phantom{;}+16\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+4;}\underline{-4x^{3}+8x^{2}-16x\phantom{;}\phantom{-;x^n}}\\\phantom{-4x^{3}+8x^{2}-16x\phantom{;};}\phantom{;}6x^{2}-15x\phantom{;}+16\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}-2x\phantom{;}+4-;x^n;}\underline{-6x^{2}+12x\phantom{;}-24\phantom{;}\phantom{;}}\\\phantom{;-6x^{2}+12x\phantom{;}-24\phantom{;}\phantom{;}-;x^n;}-3x\phantom{;}-8\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve problems step by step online. Find the integral int((4x^3-2x^2x+16)/(x^2-2x+4))dx. Divide 4x^3-2x^2+x+16 by x^2-2x+4. Resulting polynomial. Expand the integral \int\left(4x+6+\frac{-3x-8}{x^2-2x+4}\right)dx into 3 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int4xdx results in: 2x^2.

Final Answer

$2x^2+6x+3\ln\left(\frac{\sqrt{3}}{\sqrt{3+\left(x-1\right)^2}}\right)-\frac{11\sqrt{3}}{3}\arctan\left(0.577348\left(x-1\right)\right)+C_0$

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 ((4x^3+-2x^2)/(x^2+-2x))dx using partial fractionsSolve integral of ((4x^3+-2x^2)/(x^2+-2x))dx using basic integralsSolve integral of ((4x^3+-2x^2)/(x^2+-2x))dx using u-substitutionSolve integral of ((4x^3+-2x^2)/(x^2+-2x))dx using integration by partsSolve integral of ((4x^3+-2x^2)/(x^2+-2x))dx using trigonometric substitution

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Function Plot

Plotting: $2x^2+6x+3\ln\left(\frac{\sqrt{3}}{\sqrt{3+\left(x-1\right)^2}}\right)-\frac{11\sqrt{3}}{3}\arctan\left(0.577348\left(x-1\right)\right)+C_0$

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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
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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