Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following integrals:

$\int\frac{2\text{x}}{2+\text{x}-\text{x}^2}\text{ dx}$

✓

Answer

$\int\frac{2\text{x}\text{ dx}}{(2+\text{x}-\text{x}^2)}$

$2\text{x}=\text{A}\frac{\text{d}}{\text{dx}}\big(2+\text{x}-\text{x}^2\big)+\text{B}$

$2\text{x}=\text{A}(0+1-2\text{x})+\text{B}$

$2\text{x}=(-2\text{A})\text{x}+\text{A}+\text{B}$

Comparing the coefficients of like power of x,

$-2\text{A}=2$

$\text{A}=-1$

$\text{A}+\text{B}=0$

$-1+\text{B}=0$

$\text{B}=1$

Now, $\int\frac{2\text{x}\text{ dx}}{(2+\text{x}-\text{x}^2)}$

$=\int\Big(\frac{-1(1-2\text{x})+1}{-\text{x}^2+\text{x}+2}\Big)\text{dx}$

$=-\int\Big(\frac{1-2\text{x}}{-\text{x}^2+\text{x}+2}\Big)\text{dx}+\int\frac{\text{dx}}{-\text{x}^2+\text{x}+2}$

$=-\text{I}_1+\text{I}_2\ ....(1)$ (say) where

$\text{I}_1=\int\Big(\frac{1-2\text{x}}{-\text{x}^2+\text{x}+2}\Big)\text{dx}$

$\text{I}_2=\int\frac{\text{dx}}{-\text{x}^2+\text{x}+2}$

$\text{I}_1=\int\Big(\frac{1-2\text{x}}{-\text{x}^2+\text{x}+2}\Big)\text{dx}$

Let $-\text{x}^2+\text{x}+2=\text{t}$

$\Rightarrow(1-2\text{x})\text{dx}=\text{dt}$

$\text{I}_1=\int\frac{\text{dt}}{\text{t}}$

$\text{I}_1=\log|\text{t}|+\text{C}_1\ ....(2)$

$\text{I}_2=\int\frac{\text{dx}}{-\text{x}^2+\text{x}+2}$

$\text{I}_2=\int\frac{-\text{dx}}{\text{x}^2+\text{x}+2}$

$\text{I}_2=\int\frac{-\text{dx}}{\text{x}^2-\text{x}+\big(\frac{1}{2}\big)^2-\big(\frac{1}{2}\big)^2-2}$

$\text{I}_2=\int\frac{-\text{dx}}{\big(\text{x}-\frac{1}{2}\big)^2-\big(\frac{3}{2}\big)^2}$

$\text{I}_2=-\frac{1}{2\times\frac{3}{2}}\log\Bigg|\frac{\text{x}-\frac{1}{2}-\frac{3}{2}}{\text{x}-\frac{1}{2}+\frac{3}{2}}\Bigg|+\text{C}_2$

$\text{I}_2=-\frac{1}{3}\log\Big|\frac{\text{x}-2}{\text{x}+1}\Big|+\text{C}_2\ ....(3)$

From (1) (2) and (3)

$\int\Big(\frac{2\text{x}}{2+\text{x}+\text{x}^2}\Big)\text{ dx}=-\log\big|2+\text{x}-\text{x}^2\big|-\frac{1}{3}\log\Big|\frac{\text{x}-2}{\text{x}+1}\Big|+\text{C}_1+\text{C}_2$

$=-\log\big|2+\text{x}-\text{x}^2\big|+\frac{1}{3}\log\Big|\frac{1+\text{x}}{\text{x}-2}\Big|+\text{C}$

Where, $\text{C}=\text{C}_1+\text{C}_2$

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 "4 Marks" 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

Find the shortest distance between the lines $\frac{\text{x}-2}{-1}=\frac{\text{y}-5}{2}=\frac{\text{z}-0}{3}$ and $\frac{\text{x}-0}{2}=\frac{\text{y}+5}{-1}=\frac{\text{z}-1}{2}.$
→
Show that the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.
→
A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.
  1. What number of rackets and bats must be made if the factory is to work at full capacity?
  2. If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.
→
If $\log\sqrt{\text{x}^2+\text{y}^2}=\tan^{-1}\Big(\frac{\text{y}}{\text{x}}\Big),$ prove that $\frac{\text{dx}}{\text{dx}}=\frac{\text{x}+\text{y}}{\text{x}-\text{y}}$
→
Differentiate the following functions with respect to x:
$\sqrt{\tan^{-1}\big(\frac{\text{x}}{2}\big)}$
→
Prove the following results:
$2\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{8}=\tan^{-1}\frac{4}{7}$
→
R is a relation on the set Z of integers and it is given by (x, y) ∈ R ⇔ | x - y | ≤ 1. Then, R is:
  1. Reflexive and transitive.
  2. Reflexive and symmetric.
  3. Symmetric and transitive.
  4. An equivalence relation.
→
Show that the matrix $\text{A}=\begin{bmatrix}2&3\\1&2\end{bmatrix}$ satisfies the equation A3 - 4A2 + A = 0.
→
If $\text{x}=\frac{\sin^3\text{t}}{\sqrt{\cos^2\text{t}}},\text{y}=\frac{\cos^3\text{t}}{\sqrt{\cos2\text{t}}},$ find $\frac{\text{dy}}{\text{dx}}$
→
Let $\vec{\text{u}},\vec{\text{v}}$ and $\vec{\text{w}}$ be vectors such $\vec{\text{u}}+\vec{\text{v}}+\vec{\text{w}}=\vec{0}.$ If $|\vec{\text{u}}|=3,|\vec{\text{v}}|=4$ and $|\vec{\text{w}}|=5,$ then find $\vec{\text{u}}.\vec{\text{v}}+\vec{\text{v}}.\vec{\text{w}}+\vec{\text{w}}.\vec{\text{u}}.$
→