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Find the integral $\int\frac{2x^2+16x+22}{x^2+6x+10}dx$

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Solution

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Final Answer

$2x+2\ln\left(1+\left(x+3\right)^2\right)-10\arctan\left(x+3\right)+C_0$
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Step-by-step Solution

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Divide $2x^2+16x+22$ by $x^2+6x+10$

$\begin{array}{l}\phantom{\phantom{;}x^{2}+6x\phantom{;}+10;}{\phantom{;}2\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+6x\phantom{;}+10\overline{\smash{)}\phantom{;}2x^{2}+16x\phantom{;}+22\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+6x\phantom{;}+10;}\underline{-2x^{2}-12x\phantom{;}-20\phantom{;}\phantom{;}}\\\phantom{-2x^{2}-12x\phantom{;}-20\phantom{;}\phantom{;};}\phantom{;}4x\phantom{;}+2\phantom{;}\phantom{;}\\\end{array}$

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

$\begin{array}{l}\phantom{\phantom{;}x^{2}+6x\phantom{;}+10;}{\phantom{;}2\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+6x\phantom{;}+10\overline{\smash{)}\phantom{;}2x^{2}+16x\phantom{;}+22\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+6x\phantom{;}+10;}\underline{-2x^{2}-12x\phantom{;}-20\phantom{;}\phantom{;}}\\\phantom{-2x^{2}-12x\phantom{;}-20\phantom{;}\phantom{;};}\phantom{;}4x\phantom{;}+2\phantom{;}\phantom{;}\\\end{array}$

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int((2x^2+16x+22)/(x^2+6x+10))dx. Divide 2x^2+16x+22 by x^2+6x+10. Resulting polynomial. Expand the integral \int\left(2+\frac{4x+2}{x^2+6x+10}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int2dx results in: 2x.

Final Answer

$2x+2\ln\left(1+\left(x+3\right)^2\right)-10\arctan\left(x+3\right)+C_0$

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 ((2x^2+16x)/(x^2+6x))dx using partial fractionsSolve integral of ((2x^2+16x)/(x^2+6x))dx using basic integralsSolve integral of ((2x^2+16x)/(x^2+6x))dx using u-substitutionSolve integral of ((2x^2+16x)/(x^2+6x))dx using integration by partsSolve integral of ((2x^2+16x)/(x^2+6x))dx using trigonometric substitution

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Plotting: $2x+2\ln\left(1+\left(x+3\right)^2\right)-10\arctan\left(x+3\right)+C_0$

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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
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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