Integrate the rational function: x (x-1)(x-2)(x-3) - Vidyadip

Question

Integrate the rational function: $\frac{x}{(x-1)(x-2)(x-3)}$

✓

Answer

Let $\frac{x}{(x-1)(x-2)(x-3)}=\frac{A}{(x-1)}+\frac{B}{(x-2)}+\frac{C}{(x-3)}$
⇒ x = A(x - 2)(x - 3) + B(x - 1)(x - 3) + C(x - 1)(x - 2) ......(i)
Substituting x = 1, 2 and 3 respectively in equation (1), we get,
$A=\frac{1}{2}, B=-2$ and $C=\frac{3}{2}$
Thus,
$\frac{x}{(x-1)(x-2)(x-3)}=\frac{1}{2(x-1)}-\frac{2}{(x-2)}+\frac{3}{2(x-3)}$
$\Rightarrow$$\int \frac{x}{(x-1)(x-2)(x-3)} d x=\int\left\{\frac{1}{2(x-1)}-\frac{2}{(x-2)}+\frac{3}{2(x-3)}\right\} d x$
= $\frac{1}{2} \log |x-1|-2 \log |x-2|+\frac{3}{2} \log |x-3|+C$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks" from Integrals→Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

In each of the verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:
$\text{y}=\sqrt{\text{a}^2-\text{x}^2}\text{x} \in(-\text{a, a}) :\ \text{x+y}\frac{\text{dy}}{\text{dx}}=\ 0 (\text{y}\neq0)$

Evaluate the following integrals:
$\int\frac{\cos\text{x}-\sin\text{x}}{1+\sin2\text{x}}\text{dx}$

Differentiate the following functions with respect to x:
$\sin^2(2\text{x}+1)$

Express $\overrightarrow{\text{AB}}$ in terms of unit vectors $\hat{\text{i}}\text{ and }\hat{\text{j}}$, when the point is:

A(4, -1), B(1, 3)

Find $\Big|\overrightarrow{\text{AB}}\Big|$

Determine the value of $\lambda$ for which the following planes are perpendicular to other.
$2\text{x}-4\text{y}+3\text{z}=5$ and $\text{x}+2\text{y}+\lambda\text{z}=5$

Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}5&20\\0&-1 \end{vmatrix}$

Write the angle between the curves y² = 4x and x² = 2y - 3 at the point (1, 2).

Using vectors, find the value of $\lambda$ such that the points ($\lambda$, -10, 3), (1, -1, 3) and (3, 5, 3) are collinear.

Find the integral value of x, if $\begin{vmatrix}\text{x}^2&\text{x}&1\\0&2&1\\3&1&4 \end{vmatrix}=28.$

Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=[\text{x}]\text{ for }-1\leq\text{x}\leq1,$ where [x] denotes the greatest integer not exceeding x.