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

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Solution

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Final Answer

$\frac{x^{3}}{3}+2x+\ln\left(\frac{\sqrt{2}}{\sqrt{x^2+2}}\right)+2\sqrt{2}\arctan\left(0.707106x\right)+C_0$
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Step-by-step Solution

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

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

Learn how to solve integrals with radicals problems step by step online.

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

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Learn how to solve integrals with radicals problems step by step online. Find the integral int((x^5-x^2)/(x^3-2x))dx. Divide x^5-x^2 by x^3-2x. Resulting polynomial. Expand the integral \int\left(x^{2}+2+\frac{-x^{2}+4x}{x^3-2x}\right)dx into 3 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

$\frac{x^{3}}{3}+2x+\ln\left(\frac{\sqrt{2}}{\sqrt{x^2+2}}\right)+2\sqrt{2}\arctan\left(0.707106x\right)+C_0$

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

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Function Plot

Plotting: $\frac{x^{3}}{3}+2x+\ln\left(\frac{\sqrt{2}}{\sqrt{x^2+2}}\right)+2\sqrt{2}\arctan\left(0.707106x\right)+C_0$

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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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Main Topic: Integrals with Radicals

Integrals with radicals are those integrals that contain a radical (square root, cubic, etc.) in the numerator or denominator of the integral.

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