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Find the integral $\int\frac{3x^5-2x^3+5x^2-2}{x^3+1}dx$

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Solution

Final Answer

$x^{3}-2x-\frac{4}{3}\ln\left(\frac{3}{2\sqrt{\frac{9}{4}+3\left(x-\frac{1}{2}\right)^2}}\right)+\frac{2}{3}\ln\left(x+1\right)+C_0$
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Step-by-step Solution

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Divide $3x^5-2x^3+5x^2-2$ by $x^3+1$

$\begin{array}{l}\phantom{\phantom{;}x^{3}+1;}{\phantom{;}3x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{;}x^{3}+1\overline{\smash{)}\phantom{;}3x^{5}\phantom{-;x^n}-2x^{3}+5x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{3}+1;}\underline{-3x^{5}\phantom{-;x^n}\phantom{-;x^n}-3x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-3x^{5}-3x^{2};}-2x^{3}+2x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}+1-;x^n;}\underline{\phantom{;}2x^{3}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{;\phantom{;}2x^{3}+2\phantom{;}\phantom{;}-;x^n;}\phantom{;}2x^{2}\phantom{-;x^n}\phantom{-;x^n}\\\end{array}$

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$\begin{array}{l}\phantom{\phantom{;}x^{3}+1;}{\phantom{;}3x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{;}x^{3}+1\overline{\smash{)}\phantom{;}3x^{5}\phantom{-;x^n}-2x^{3}+5x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{3}+1;}\underline{-3x^{5}\phantom{-;x^n}\phantom{-;x^n}-3x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-3x^{5}-3x^{2};}-2x^{3}+2x^{2}\phantom{-;x^n}-2\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}+1-;x^n;}\underline{\phantom{;}2x^{3}\phantom{-;x^n}\phantom{-;x^n}+2\phantom{;}\phantom{;}}\\\phantom{;\phantom{;}2x^{3}+2\phantom{;}\phantom{;}-;x^n;}\phantom{;}2x^{2}\phantom{-;x^n}\phantom{-;x^n}\\\end{array}$

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Learn how to solve problems step by step online. Find the integral int((3x^5-2x^35x^2+-2)/(x^3+1))dx. Divide 3x^5-2x^3+5x^2-2 by x^3+1. Resulting polynomial. Simplify the expression inside the integral. The integral \int3x^{2}dx results in: x^{3}.

Final Answer

$x^{3}-2x-\frac{4}{3}\ln\left(\frac{3}{2\sqrt{\frac{9}{4}+3\left(x-\frac{1}{2}\right)^2}}\right)+\frac{2}{3}\ln\left(x+1\right)+C_0$

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^5+-2x^3)/(x^3+1))dx using partial fractionsSolve integral of ((3x^5+-2x^3)/(x^3+1))dx using basic integralsSolve integral of ((3x^5+-2x^3)/(x^3+1))dx using u-substitutionSolve integral of ((3x^5+-2x^3)/(x^3+1))dx using integration by partsSolve integral of ((3x^5+-2x^3)/(x^3+1))dx using trigonometric substitution

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Function Plot

Plotting: $x^{3}-2x-\frac{4}{3}\ln\left(\frac{3}{2\sqrt{\frac{9}{4}+3\left(x-\frac{1}{2}\right)^2}}\right)+\frac{2}{3}\ln\left(x+1\right)+C_0$

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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
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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