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Find the integral $\int\frac{6x^4+23x^2+4x+5}{x^2+3}dx$

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Solution

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Final Answer

$2x^{3}+5x-\frac{10\sqrt{3}}{3}\arctan\left(0.577348x\right)-4\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$
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Step-by-step Solution

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Divide $6x^4+23x^2+4x+5$ by $x^2+3$

$\begin{array}{l}\phantom{\phantom{;}x^{2}+3;}{\phantom{;}6x^{2}\phantom{-;x^n}+5\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3\overline{\smash{)}\phantom{;}6x^{4}\phantom{-;x^n}+23x^{2}+4x\phantom{;}+5\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3;}\underline{-6x^{4}\phantom{-;x^n}-18x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-6x^{4}-18x^{2};}\phantom{;}5x^{2}+4x\phantom{;}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3-;x^n;}\underline{-5x^{2}\phantom{-;x^n}-15\phantom{;}\phantom{;}}\\\phantom{;-5x^{2}-15\phantom{;}\phantom{;}-;x^n;}\phantom{;}4x\phantom{;}-10\phantom{;}\phantom{;}\\\end{array}$

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

$\begin{array}{l}\phantom{\phantom{;}x^{2}+3;}{\phantom{;}6x^{2}\phantom{-;x^n}+5\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+3\overline{\smash{)}\phantom{;}6x^{4}\phantom{-;x^n}+23x^{2}+4x\phantom{;}+5\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+3;}\underline{-6x^{4}\phantom{-;x^n}-18x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-6x^{4}-18x^{2};}\phantom{;}5x^{2}+4x\phantom{;}+5\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+3-;x^n;}\underline{-5x^{2}\phantom{-;x^n}-15\phantom{;}\phantom{;}}\\\phantom{;-5x^{2}-15\phantom{;}\phantom{;}-;x^n;}\phantom{;}4x\phantom{;}-10\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int((6x^4+23x^24x+5)/(x^2+3))dx. Divide 6x^4+23x^2+4x+5 by x^2+3. Resulting polynomial. Expand the integral \int\left(6x^{2}+5+\frac{4x-10}{x^2+3}\right)dx into 3 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int6x^{2}dx results in: 2x^{3}.

Final Answer

$2x^{3}+5x-\frac{10\sqrt{3}}{3}\arctan\left(0.577348x\right)-4\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$

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 ((6x^4+23x^2)/(x^2+3))dx using partial fractionsSolve integral of ((6x^4+23x^2)/(x^2+3))dx using basic integralsSolve integral of ((6x^4+23x^2)/(x^2+3))dx using u-substitutionSolve integral of ((6x^4+23x^2)/(x^2+3))dx using integration by partsSolve integral of ((6x^4+23x^2)/(x^2+3))dx using trigonometric substitution

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Function Plot

Plotting: $2x^{3}+5x-\frac{10\sqrt{3}}{3}\arctan\left(0.577348x\right)-4\ln\left(\frac{\sqrt{3}}{\sqrt{x^2+3}}\right)+C_0$

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7
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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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Main Topic: Integrals of Rational Functions

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

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