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Find the integral $\int\frac{\left(x^2+1\right)^2+5}{x^2+1}dx$

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Solution

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Final Answer

$\frac{x^{3}}{3}+x+5\arctan\left(x\right)+C_0$
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Step-by-step Solution

Problem to solve:

$\int\frac{\left(x^2+1\right)^2+5}{x^2+1}dx$

Specify the solving method

1

Expand the fraction $\frac{\left(x^2+1\right)^2+5}{x^2+1}$ into $2$ simpler fractions with common denominator $x^2+1$

$\int\left(\frac{\left(x^2+1\right)^2}{x^2+1}+\frac{5}{x^2+1}\right)dx$

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

$\int\left(\frac{\left(x^2+1\right)^2}{x^2+1}+\frac{5}{x^2+1}\right)dx$

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int(((x^2+1)^2+5)/(x^2+1))dx. Expand the fraction \frac{\left(x^2+1\right)^2+5}{x^2+1} into 2 simpler fractions with common denominator x^2+1. Simplify the resulting fractions. Expand the integral \int\left(x^2+1+\frac{5}{x^2+1}\right)dx into 3 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int x^2dx results in: \frac{x^{3}}{3}.

Final Answer

$\frac{x^{3}}{3}+x+5\arctan\left(x\right)+C_0$

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 int(((x^2+1)^2+5)/(x^2+1))dx using partial fractionsSolve int(((x^2+1)^2+5)/(x^2+1))dx using basic integralsSolve int(((x^2+1)^2+5)/(x^2+1))dx using u-substitutionSolve int(((x^2+1)^2+5)/(x^2+1))dx using integration by partsSolve int(((x^2+1)^2+5)/(x^2+1))dx using trigonometric substitution

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(◻)
+
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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

How to improve your answer:

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Main topic:

Integrals of Rational Functions

Used formulas:

3. See formulas

Time to solve it:

~ 0.08 s

Related topics:

Integrals of Rational FunctionsIntegral CalculusCalculusIndefinite Integrals

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