Integrate the following function:

$\frac{x}{(x^2 + 1)(x - 1)}$

Question

Integrate the following function:

$\frac{x}{(x^2 + 1)(x - 1)}$

Accepted Answer

> [!TLDR]
> With the irreducible factor $x^2+1$ carrying an $Ax+B$ numerator, decompose and integrate to $-\frac14\log(x^2+1) + \frac12	an^{-1}x + \frac12\log|x-1| + C$.

## Setting up the form

An irreducible quadratic factor needs a **linear** numerator; a linear factor needs a constant:
$$\frac{x}{(x^2+1)(x-1)} = \frac{Ax+B}{x^2+1} + \frac{C}{x-1}.$$

## Step 1 — clear denominators and solve

$$x = (Ax+B)(x-1) + C(x^2+1) = (A+C)x^2 + (B-A)x + (C-B).$$
Matching $x^2$, $x^1$, $x^0$ against $0\cdot x^2 + 1\cdot x + 0$:
$$A+C=0,\qquad B-A=1,\qquad C-B=0.$$
From the last, $C=B$; from the first, $A=-C=-B$; then $B-A = 2B = 1 \Rightarrow B=	frac12$, so $A=-	frac12$, $C=	frac12$.

## Step 2 — split the quadratic piece

$$\frac{-	frac12 x + 	frac12}{x^2+1} = -\frac12\cdot\frac{x}{x^2+1} + \frac12\cdot\frac{1}{x^2+1}.$$

## Step 3 — integrate each term

- $\displaystyle -\frac12\int\frac{x}{x^2+1}\,dx = -\frac14\log(x^2+1)$ (let $u=x^2+1$). …

Q.Integrate the following function: $\frac{x}{(x^2 + 1)(x - 1)}$

Concept understanding — Partial Fraction Decomposition

Partial Fraction Decomposition

When it applies

The standard forms

Worked example

Setting up the form

Step 1 — clear denominators and solve

Step 2 — split the quadratic piece

Step 3 — integrate each term

Method: Partial fractions with an irreducible quadratic factor

Steps

Common Mistakes

Q.Integrate the following function: x(x2+1)(x−1)\frac{x}{(x^2 + 1)(x - 1)}(x2+1)(x−1)x​

Setting up the form

Step 1 — clear denominators and solve

Step 2 — split the quadratic piece

Step 3 — integrate each term

🎓Unlock everything free for 14 days

Q.Integrate the following function: $\frac{x}{(x^2 + 1)(x - 1)}$

Unlock everything free for 14 days