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Find the integral $\int\frac{2x^5+1}{x^3-4x^2-5x}dx$

Step-by-step Solution

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Final Answer

$\frac{2}{3}x^{3}+4x^2+42x-\frac{1}{6}\ln\left(x+1\right)+208.366667\ln\left(x-5\right)-\frac{1}{5}\ln\left(x\right)+C_0$
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Step-by-step Solution

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We could not solve this problem by using the method: Integrate by trigonometric substitution

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Divide $2x^5+1$ by $x^3-4x^2-5x$

$\begin{array}{l}\phantom{\phantom{;}x^{3}-4x^{2}-5x\phantom{;};}{\phantom{;}2x^{2}+8x\phantom{;}+42\phantom{;}\phantom{;}}\\\phantom{;}x^{3}-4x^{2}-5x\phantom{;}\overline{\smash{)}\phantom{;}2x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{3}-4x^{2}-5x\phantom{;};}\underline{-2x^{5}+8x^{4}+10x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-2x^{5}+8x^{4}+10x^{3};}\phantom{;}8x^{4}+10x^{3}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-4x^{2}-5x\phantom{;}-;x^n;}\underline{-8x^{4}+32x^{3}+40x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-8x^{4}+32x^{3}+40x^{2}-;x^n;}\phantom{;}42x^{3}+40x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-4x^{2}-5x\phantom{;}-;x^n-;x^n;}\underline{-42x^{3}+168x^{2}+210x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-42x^{3}+168x^{2}+210x\phantom{;}-;x^n-;x^n;}\phantom{;}208x^{2}+210x\phantom{;}+1\phantom{;}\phantom{;}\\\end{array}$

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$\begin{array}{l}\phantom{\phantom{;}x^{3}-4x^{2}-5x\phantom{;};}{\phantom{;}2x^{2}+8x\phantom{;}+42\phantom{;}\phantom{;}}\\\phantom{;}x^{3}-4x^{2}-5x\phantom{;}\overline{\smash{)}\phantom{;}2x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{3}-4x^{2}-5x\phantom{;};}\underline{-2x^{5}+8x^{4}+10x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-2x^{5}+8x^{4}+10x^{3};}\phantom{;}8x^{4}+10x^{3}\phantom{-;x^n}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-4x^{2}-5x\phantom{;}-;x^n;}\underline{-8x^{4}+32x^{3}+40x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;-8x^{4}+32x^{3}+40x^{2}-;x^n;}\phantom{;}42x^{3}+40x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{3}-4x^{2}-5x\phantom{;}-;x^n-;x^n;}\underline{-42x^{3}+168x^{2}+210x\phantom{;}\phantom{-;x^n}}\\\phantom{;;-42x^{3}+168x^{2}+210x\phantom{;}-;x^n-;x^n;}\phantom{;}208x^{2}+210x\phantom{;}+1\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve problems step by step online. Find the integral int((2x^5+1)/(x^3-4x^2-5x))dx. Divide 2x^5+1 by x^3-4x^2-5x. Resulting polynomial. Expand the integral \int\left(2x^{2}+8x+42+\frac{208x^{2}+210x+1}{x^3-4x^2-5x}\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int2x^{2}dx results in: \frac{2}{3}x^{3}.

Final Answer

$\frac{2}{3}x^{3}+4x^2+42x-\frac{1}{6}\ln\left(x+1\right)+208.366667\ln\left(x-5\right)-\frac{1}{5}\ln\left(x\right)+C_0$

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 ((2x^5+1)/(x^3+-4x^2))dx using partial fractionsSolve integral of ((2x^5+1)/(x^3+-4x^2))dx using basic integralsSolve integral of ((2x^5+1)/(x^3+-4x^2))dx using u-substitutionSolve integral of ((2x^5+1)/(x^3+-4x^2))dx using integration by parts

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Function Plot

Plotting: $\frac{2}{3}x^{3}+4x^2+42x-\frac{1}{6}\ln\left(x+1\right)+208.366667\ln\left(x-5\right)-\frac{1}{5}\ln\left(x\right)+C_0$

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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
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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