Evaluate the following intregals: 1 (x-1)(x+1)(x+2) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{1}{(\text{x}-1)(\text{x}+1)(\text{x}+2)}\ \text{dx}$

✓

Answer

Let $\text{I}=\int\frac{1}{(\text{x}-1)(\text{x}+1)(\text{x}+2)}=\frac{\text{A}}{\text{x}-1}+\frac{\text{B}}{\text{x}+1}+\frac{\text{C}}{\text{x}+2}$
$\Rightarrow1=\text{A}(\text{x}+1)(\text{x}+2)+\text{B}(\text{x}-1)(\text{x}+2)+\text{C}(\text{x}^2-1)$
Put x = 1
$\Rightarrow1=6\text{A}\Rightarrow\text{A}=\frac{1}{6}$
put = -1
$\Rightarrow1=-2\text{B}\Rightarrow\text{B}=-\frac{1}{2}$
put = -2
$\Rightarrow1=3\text{C}\Rightarrow\text{C}=\frac{1}{3}$
So,
$\text{I}=\frac{1}{6}\int\frac{\text{dx}}{\text{x}-1}-\frac{1}{2}\int\frac{\text{dx}}{\text{x}+1}+\frac{1}{3}\int\frac{\text{dx}}{\text{x}+2}$
$\text{I}=\frac{1}{6}\log|\text{x}-1|-\frac{1}{2}\log|\text{x}+1|+\frac{1}{3}\log|\text{x}+2|+\text{C}$

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 Question" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The probability that at least one of the two events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, evaluate $\text{P}\bar{(\text{A})}+\text{P}\bar{(\text{B})}.$

A dice is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

On the set Z of integers a binary operation * is defined by a * b = ab + 1 for all a, b ∈ Z. Prove that * is not associative on Z.

The volume of a cube is increasing at the rate of $8\ cm^3/sec.$ How fast is the surface area increasing when the length of an edge is $12\ cm?$

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{y}=\sqrt{\text{x}^3}\ \text{at}\text{ x}=4$

Find the mean and standard deviation of the following probability distributions:

$\text{x}_\text{i}$	$-5$	$-4$	$1$	$2$
$\text{p}_\text{i}$	$\frac{1}{4}$	$\frac{1}{8}$	$\frac{1}{2}$	$\frac{1}{8}$

If the vertices A, B, C of a triangle ABC are the points with position vectors $\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},\ \text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}},\ \text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}$ respectively, what are the vectors determined by its sides? Find the length of these vectors.

If $|\vec{\text{a}}|=2,\big|\vec{\text{b}}\big|=5$ and $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=8,$ find $\vec{\text{a}}.\vec{\text{b}}.$

If $\text{A}=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\text{ and A (adj A =)}\begin{bmatrix} \text{k} & 0 \\ 0 & \text{k} \end{bmatrix},$ then find the value of k.

Show that $\begin{vmatrix}\sin10^\circ&-\cos10^\circ\\\sin80^\circ&\cos80^\circ \end{vmatrix}=1$