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

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Solution

Final Answer

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

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

$\begin{array}{l}\phantom{\phantom{;}x^{4}+x^{3};}{\phantom{;}x\phantom{;}\phantom{-;x^n}}\\\phantom{;}x^{4}+x^{3}\overline{\smash{)}\phantom{;}x^{5}+x^{4}\phantom{-;x^n}-2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{4}+x^{3};}\underline{-x^{5}-x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}-x^{4};}-2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\end{array}$

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$\begin{array}{l}\phantom{\phantom{;}x^{4}+x^{3};}{\phantom{;}x\phantom{;}\phantom{-;x^n}}\\\phantom{;}x^{4}+x^{3}\overline{\smash{)}\phantom{;}x^{5}+x^{4}\phantom{-;x^n}-2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{4}+x^{3};}\underline{-x^{5}-x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{5}-x^{4};}-2x^{2}\phantom{-;x^n}+1\phantom{;}\phantom{;}\\\end{array}$

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

Final Answer

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

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+x^4)/(x^4+x^3))dx using partial fractionsSolve integral of ((x^5+x^4)/(x^4+x^3))dx using basic integralsSolve integral of ((x^5+x^4)/(x^4+x^3))dx using u-substitutionSolve integral of ((x^5+x^4)/(x^4+x^3))dx using integration by partsSolve integral of ((x^5+x^4)/(x^4+x^3))dx using trigonometric substitution

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Function Plot

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

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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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