Download App

Indefinite Integrals — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIndefinite Integrals5 Marks
Question
Evaluvate the following intregals:
$\int\frac{2\text{x}+1}{(\text{x}+1)(\text{x}-2)}\ \text{dx}$
✓

Answer

Let $\int\frac{2\text{x}+1}{(\text{x}+1)(\text{x}-2)}=\frac{\text{A}}{(\text{x}+1)}+\frac{\text{B}}{(\text{x}-2)}$$\Rightarrow2\text{x}+1=\text{A}(\text{x}-2)+\text{B}(\text{x}+1)$
Put $\text{x}=2$
$\Rightarrow5=3\text{B}\Rightarrow\text{B}=\frac{5}{3}$
Put $\text{x}=-1$
$\Rightarrow-1=-3\text{A}\Rightarrow\text{A}=\frac{1}{3}$
so,
$\int\frac{2\text{x}+1}{(\text{x}+1)(\text{x}-2)}\text{ dx}=\frac{1}{3}\int\frac{\text{dx}}{\text{x}+1}+\frac{5}{3}\int\frac{\text{dx}}{\text{x}-2}$
$=\frac{1}{3}\log|\text{x}+1|+\frac{5}{3}\log|\text{x}-2|+\text{C}$
Thus
$\text{I}=\frac{1}{3}\log|\text{x}+1|+\frac{5}{3}\log|\text{x}-2|+\text{C}$

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 "5 Marks Questions" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the foot of the perpendicular drawn from the point $\hat{\text{i}}+6\hat{\text{j}}+3\hat{\text{k}}$ to the line $\vec{\text{r}}=\hat{\text{j}}+2\hat{\text{k}}+\lambda\big(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}\big).$ Also, find the length of the perpendicylar
→
$\text{If A} = \begin{bmatrix} 1 & -2 & 3 \\ 0 & -1 & 4 \\ -2 & 2 & 1 \end{bmatrix}, \text{find (A')}^{-1}. $
→
Find the area of the bounded by the curve xy - 3x - 2y = 0, x-axis and the lins x = 3, x = 4.
→
Find the value of 'a' for which the function f defined as$ \begin{matrix} & \text{a sin}\frac{\pi}{2}\text{(x + 1)}, & x\leq0 \\ \text{f(x)} \\ & \frac{\text{tan x - sin x}}{\text{x}^{3}}, & x<0 \\ \end{matrix}$
is continuous at X = 0.
→
An item is manufactured by three machines $A, B$ and $C$. Out of the total number of items manufactured during a specified period, $50\%$ are manufactured on $A, 30\%$ on $B$ and $20\%$ on $C. 2\%$ of the items produced on $A$ and $2\%$ of items produced on $B$ are defective, and $3\%$ of these produced on $C$ are defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine A?
→
Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=(\lambda-1)\hat{\text{i}}+(\lambda+1)\hat{\text{j}}-(1+\lambda)\hat{\text{k}}$ and $\vec{\text{r}}=(1-\mu)\hat{\text{i}}+(2\mu-1)\hat{\text{j}}+(\mu+2)\hat{\text{k}}$
→
A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:
 
 
Product A
Product B
Weekly capacity
Department 1
3
2
130
Department 2
4
6
260
Selling price per unit
Rs. 25
Rs. 30
 
Labour cost per unit
Rs. 16
Rs. 20
 
Raw material cost per unit
Rs. 4
Rs. 4
 
The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.
→
Integrate the rational function $\frac{x}{\left(x^{2}+1\right)(x-1)}$
→
If $\text{y}=\log\sqrt{\frac{1+\tan\text{x}}{1-\tan\text{x}}},$ prove that $\frac{\text{dy}}{\text{dx}}=\sec2\text{x}$
→
Evaluate the following integrals:
$\int\limits^\pi_0\Big(\frac{\text{x}}{1+\sin^2\text{x}}+\cos^7\text{x}\Big)\text{dx}$
→