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Find the integral $\int\frac{x^4+x+1}{x^2+8x-2}dx$

Used Formulas

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Solution

Formulas

Videos

Basic Integrals

· Sum Rule for Integration
$\int\left(a+b+...\right)dx=\int adx+\int bdx+...$
· Power Rule of Integration
$\int x^ndx=\frac{x^{\left(n+1\right)}}{n+1}+C$
· Constant factor Rule
$\int cxdx=c\int xdx$
· Integral of a Constant
$\int cdx=cvar+C$

Integration Techniques

· Integration by Substitution
$\int f\left(x\right)dx=\int f\left(g\left(t\right)\right) g'\left(t\right)dt$
· Integration by Parts
$\int udv=uv - \int vdu$

Basic Derivatives

· Derivative of the linear function
$\frac{d}{dx}\left(x\right)=1$

Special Products

· Difference of two Squares
$a^2-b^2=(a+b)(a-b)$

Integrals of Rational Functions

$\int\frac{n}{x+b}dx=nsign\left(x\right)\ln\left(x+b\right)+C$
$\int\frac{n}{a+b}dx=n\int\frac{1}{a+b}dx$

Function Plot

Plotting: $\frac{x^{3}}{3}-4x^2+66x-543\left(\left(-\frac{1}{6\sqrt{2}}\ln\left(8.2426407+x\right)+\frac{1}{6\sqrt{2}}\ln\left(-\frac{33}{136}+x\right)\right)\left(x+4\right)+\frac{1}{6\sqrt{2}}\left(-8.2426407+\left(8.2426407+x\right)\ln\left(8.2426407+x\right)-x\right)-\frac{1}{6\sqrt{2}}\left(\frac{33}{136}+\left(-\frac{33}{136}+x\right)\ln\left(-\frac{33}{136}+x\right)-x\right)\right)+271.6468551\ln\left(-\frac{33}{136}+x\right)-271.6468551\ln\left(8.2426407+x\right)+C_0$

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◻/◻
/
÷
◻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 of Rational Functions

Integrals of rational functions of the form R(x) = P(x)/Q(x).

Used Formulas

See formulas (10)

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