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Find the integral $\int\frac{2x+1}{\left(x+1\right)^3\left(x^2+4\right)^2}dx$

Step-by-step Solution

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Solution

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Final Answer

$\frac{1}{50\left(x+1\right)^{2}}-\frac{1}{322}\arctan\left(\frac{x}{2}\right)+\frac{-\frac{8}{421}x}{x^2+4}+\frac{\frac{21}{250}}{x^2+4}+\frac{38}{625}\ln\left(x+1\right)+\frac{-\frac{6}{125}}{x+1}+\frac{38}{625}\ln\left(\frac{2}{\sqrt{x^2+4}}\right)+C_0$
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Step-by-step Solution

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Rewrite the fraction $\frac{2x+1}{\left(x+1\right)^3\left(x^2+4\right)^2}$ in $5$ simpler fractions using partial fraction decomposition

$\frac{2x+1}{\left(x+1\right)^3\left(x^2+4\right)^2}=\frac{A}{\left(x+1\right)^3}+\frac{Bx+C}{\left(x^2+4\right)^2}+\frac{D}{x+1}+\frac{F}{\left(x+1\right)^{2}}+\frac{Gx+H}{x^2+4}$

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$\frac{2x+1}{\left(x+1\right)^3\left(x^2+4\right)^2}=\frac{A}{\left(x+1\right)^3}+\frac{Bx+C}{\left(x^2+4\right)^2}+\frac{D}{x+1}+\frac{F}{\left(x+1\right)^{2}}+\frac{Gx+H}{x^2+4}$

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Learn how to solve integrals of rational functions problems step by step online. Find the integral int((2x+1)/((x+1)^3(x^2+4)^2))dx. Rewrite the fraction \frac{2x+1}{\left(x+1\right)^3\left(x^2+4\right)^2} in 5 simpler fractions using partial fraction decomposition. Find the values for the unknown coefficients: A, B, C, D, F, G, H. The first step is to multiply both sides of the equation from the previous step by \left(x+1\right)^3\left(x^2+4\right)^2. Multiplying polynomials. Simplifying.

Final Answer

$\frac{1}{50\left(x+1\right)^{2}}-\frac{1}{322}\arctan\left(\frac{x}{2}\right)+\frac{-\frac{8}{421}x}{x^2+4}+\frac{\frac{21}{250}}{x^2+4}+\frac{38}{625}\ln\left(x+1\right)+\frac{-\frac{6}{125}}{x+1}+\frac{38}{625}\ln\left(\frac{2}{\sqrt{x^2+4}}\right)+C_0$

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Solve integral of ((2x+1)/(x+1)^3(x^2+4)^2)dx using partial fractionsSolve integral of ((2x+1)/(x+1)^3(x^2+4)^2)dx using basic integralsSolve integral of ((2x+1)/(x+1)^3(x^2+4)^2)dx using u-substitutionSolve integral of ((2x+1)/(x+1)^3(x^2+4)^2)dx using integration by partsSolve integral of ((2x+1)/(x+1)^3(x^2+4)^2)dx using trigonometric substitution

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Function Plot

Plotting: $\frac{1}{50\left(x+1\right)^{2}}-\frac{1}{322}\arctan\left(\frac{x}{2}\right)+\frac{-\frac{8}{421}x}{x^2+4}+\frac{\frac{21}{250}}{x^2+4}+\frac{38}{625}\ln\left(x+1\right)+\frac{-\frac{6}{125}}{x+1}+\frac{38}{625}\ln\left(\frac{2}{\sqrt{x^2+4}}\right)+C_0$

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>=
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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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