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

Step-by-step Solution

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Final Answer

$\frac{1}{2}x^2+\frac{1}{3}\ln\left(\frac{\frac{\sqrt{3}}{2}}{\sqrt{\frac{3}{4}+\left(x-\frac{1}{2}\right)^2}}\right)-\frac{\sqrt{3}}{3}\arctan\left(1.154696\left(x-\frac{1}{2}\right)\right)+\frac{1}{3}\ln\left(x+1\right)+C_0$
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Step-by-step Solution

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Divide $x^4$ by $x^3+1$

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

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

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Learn how to solve quotient rule of differentiation problems step by step online. Find the integral int((x^4)/(x^3+1))dx. Divide x^4 by x^3+1. Resulting polynomial. Simplify the expression inside the integral. The integral \int xdx results in: \frac{1}{2}x^2.

Final Answer

$\frac{1}{2}x^2+\frac{1}{3}\ln\left(\frac{\frac{\sqrt{3}}{2}}{\sqrt{\frac{3}{4}+\left(x-\frac{1}{2}\right)^2}}\right)-\frac{\sqrt{3}}{3}\arctan\left(1.154696\left(x-\frac{1}{2}\right)\right)+\frac{1}{3}\ln\left(x+1\right)+C_0$

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Function Plot

Plotting: $\frac{1}{2}x^2+\frac{1}{3}\ln\left(\frac{\frac{\sqrt{3}}{2}}{\sqrt{\frac{3}{4}+\left(x-\frac{1}{2}\right)^2}}\right)-\frac{\sqrt{3}}{3}\arctan\left(1.154696\left(x-\frac{1}{2}\right)\right)+\frac{1}{3}\ln\left(x+1\right)+C_0$

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4
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
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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