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

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Solution

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Final Answer

$\frac{1}{2}x^2+\frac{7}{4}\ln\left(x^2-4\right)-\frac{3}{2}\ln\left(\frac{x+2}{\sqrt{x^2-4}}\right)+\frac{1}{2}\ln\left(x\right)+C_0$
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Step-by-step Solution

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

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

Learn how to solve integrals of rational functions problems step by step online.

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

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

$\frac{1}{2}x^2+\frac{7}{4}\ln\left(x^2-4\right)-\frac{3}{2}\ln\left(\frac{x+2}{\sqrt{x^2-4}}\right)+\frac{1}{2}\ln\left(x\right)+C_0$

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

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Function Plot

Plotting: $\frac{1}{2}x^2+\frac{7}{4}\ln\left(x^2-4\right)-\frac{3}{2}\ln\left(\frac{x+2}{\sqrt{x^2-4}}\right)+\frac{1}{2}\ln\left(x\right)+C_0$

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◻/◻
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√

◻√◻
◻√
∞
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).

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