Integrate the function $\frac{5x}{(x+1)(x^2+9)}$

> [!TLDR] > Decompose into $\frac{A}{x+1}+\frac{Bx+C}{x^2+9}$ with $A=-\tfrac12,\,B=\tfrac12,\,C=\tfrac92$, then integrate to get $-\tfrac12\log|x+1|+\tfrac14\log(x^2+9)+\tfrac32\arctan\tfrac x3+C$. ## Set-up The factor $x+1$ is linear and $x^2+9$ is irreducible (no real roots), so the irreducible quadratic gets a **linear** numerator: $$\frac{5x}{(x+1)(x^2+9)}=\frac{A}{x+1}+\frac{Bx+C}{x^2+9}.$$ ## 1. Solve for the constants Multiply through by $(x+1)(x^2+9)$: $$5x=A(x^2+9)+(Bx+C)(x+1).$$ - $x=-1$: $-5=A(1+9)=10A\Rightarrow A=-\tfrac12$. - Coefficient of $x^2$: $A+B=0\Rightarrow B=\tfrac12$. - Constant term: $9A+C=0\Rightarrow C=\tfrac92$. - Check coefficient of $x$: $B+C=\tfrac12+\tfrac92=5$ $\checkmark$. So $$\frac{5x}{(x+1)(x^2+9)}=-\frac{1}{2(x+1)}+\frac{\tfrac12x+\tfrac92}{x^2+9}.$$ ## 2. Integrate each piece $$-\frac12\int\frac{dx}{x+1}=-\frac12\log|x+1|.$$ For the quadratic part, split $\tfrac12x+\tfrac92$: …

Integrate the function (5x)/((x+1)(x²+9))

Miscellaneous Exercise · Q6

\frac{5x}{(x+1)(x^2+9)}

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Concept understanding — Partial Fraction Decomposition

Partial Fraction Decomposition

Adding $\frac{2}{x-1}+\frac{3}{x+2}$ over a common denominator gives $\frac{5x+1}{x^2+x-2}$ . Partial fraction decomposition reverses this — it breaks one complicated rational function back into a sum of simple pieces. Those pieces are far easier to integrate: $\int \frac{2}{x-1}\,dx = 2\log|x-1|$ is immediate, while the combined fraction is not.

When it applies

You need a proper rational function, $\deg P < \deg Q$ . If the numerator's degree is equal or higher, first do polynomial long division. The denominator $Q(x)$ must be factored into linear and/or irreducible quadratic factors.

The standard forms

For $\frac{P(x)}{Q(x)}$ with $Q$ fully factored, each factor contributes a term:

Distinct linear $(ax+b)$ $\to$ $\dfrac{A}{ax+b}$ .
Repeated linear $(ax+b)^n$ $\to$ $\dfrac{A_1}{ax+b}+\dfrac{A_2}{(ax+b)^2}+\cdots+\dfrac{A_n}{(ax+b)^n}$ .
Irreducible quadratic $(ax^2+bx+c)$ $\to$ $\dfrac{Ax+B}{ax^2+bx+c}$ — a linear numerator, not just a constant.

The unknown constants are found from the resulting equations; the decomposition is unique, which is what lets us solve for them systematically.

Worked example

Decompose $\frac{3x+5}{x^2-3x+2}$ . Factor the denominator: $(x-1)(x-2)$ . Set

\frac{3x+5}{(x-1)(x-2)} = \frac{A}{x-1}+\frac{B}{x-2}.

Clear denominators: $3x+5 = A(x-2)+B(x-1)$ . Substituting the roots, $x=1$ gives $8=-A$ so $A=-8$ , and $x=2$ gives $11=B$ . Hence

\frac{3x+5}{x^2-3x+2} = \frac{-8}{x-1}+\frac{11}{x-2}.

Tip

With distinct linear factors, substitute each factor's root to knock out all but one term — much faster than equating coefficients. …

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Decompose into $\frac{A}{x+1}+\frac{Bx+C}{x^2+9}$ with $A=-\tfrac12,\,B=\tfrac12,\,C=\tfrac92$ , then integrate to get $-\tfrac12\log|x+1|+\tfrac14\log(x^2+9)+\tfrac32\arctan\tfrac x3+C$ .

Set-up

The factor $x+1$ is linear and $x^2+9$ is irreducible (no real roots), so the irreducible quadratic gets a linear numerator:

\frac{5x}{(x+1)(x^2+9)}=\frac{A}{x+1}+\frac{Bx+C}{x^2+9}.

1. Solve for the constants

Multiply through by $(x+1)(x^2+9)$ :

5x=A(x^2+9)+(Bx+C)(x+1).

$x=-1$ : $-5=A(1+9)=10A\Rightarrow A=-\tfrac12$ .
Coefficient of $x^2$ : $A+B=0\Rightarrow B=\tfrac12$ .
Constant term: $9A+C=0\Rightarrow C=\tfrac92$ .
Check coefficient of $x$ : $B+C=\tfrac12+\tfrac92=5$ $\checkmark$ .

\frac{5x}{(x+1)(x^2+9)}=-\frac{1}{2(x+1)}+\frac{\tfrac12x+\tfrac92}{x^2+9}.

2. Integrate each piece

-\frac12\int\frac{dx}{x+1}=-\frac12\log|x+1|.

For the quadratic part, split $\tfrac12x+\tfrac92$ : …

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Q.Integrate the function $\frac{5x}{(x+1)(x^2+9)}$

Concept understanding — Partial Fraction Decomposition

Partial Fraction Decomposition

When it applies

The standard forms

Worked example

Set-up

1. Solve for the constants

2. Integrate each piece

Method: Partial fractions with an irreducible quadratic factor

Steps

Common Mistakes

Q.Integrate the function 5x(x+1)(x2+9)\frac{5x}{(x+1)(x^2+9)}(x+1)(x2+9)5x​

Set-up

1. Solve for the constants

2. Integrate each piece

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Q.Integrate the function $\frac{5x}{(x+1)(x^2+9)}$

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