Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra 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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the area of the region bounded by the parabola $y^2 = 2x + 1$ and the line $x - y - 1 = 0.$
→
If $\triangle=\begin{vmatrix}1&\text{x}&\text{x}^2\\1&\text{y}&\text{y}^2\\1&\text{z}&\text{z}^2\end{vmatrix},$ $\triangle_1=\begin{vmatrix}1&1&1\\\text{yz}&\text{zx}&\text{xy}\\\text{x}&\text{y}&\text{z}\end{vmatrix},$ then prove that $\triangle+\triangle_1=0$
→
Using differentials, find the approximate values of the following:
$(0.009)^{\frac{1}{3}}$
→
If $\text{y}=\text{e}^{\text{x}^{\text{e}^\text{x}}}+\text{x}^{\text{e}^{\text{e}^\text{x}}}+\text{e}^{\text{x}^{\text{x}^{\text{e}}}},$ prove that $\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x}^{\text{e}^\text{x}}}\times\text{x}^{\text{e}^{\text{x}}}\Big\{\frac{\text{e}^\text{x}}{\text{x}}+\text{e}^\text{x}\log\text{x}\Big\}+\text{e}^{\text{x}^{\text{e}^{\text{x}}}}\times\text{e}^{\text{e}^\text{x}}\Big\{\frac{1}{\text{x}}+\text{e}^\text{x}\times\log\text{x}\Big\}+\text{e}^{\text{x}^{\text{x}^\text{e}}}\text{x}^{\text{x}^{\text{e}}}\times\text{x}^{\text{e}-1}\Big\{\text{x}+\text{e}\log\text{x}\Big\}$
→
If $\text{y}=\tan^{-1}\Big(\frac{1-\text{x}}{1+\text{x}}\Big),$, find $\frac{\text{dy}}{\text{dx}}.$
→
Find the angle between the pairs of lines with direction ratios proportional to1, $2, -2$ and $-2, 2, 1$
→
Differentiate $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big)$ with respect to $\sec^{-1}\Big(\frac{1}{\sqrt{1+\text{x}^2}}\Big),$ if:
$\text{x}\in\Big(0,\frac{1}{\sqrt{2}}\Big)$
→
Write the values of 'a' for which the following distribution of probabilities becomes a probability distributioin:
$\text{X}=\text{x}_\text{i}:$ $-2$ $-1$ $0$ $1$
$\text{P}(\text{X}=\text{x}_\text{i}):$ $\frac{1-\text{a}}{4}$ $\frac{1+2\text{a}}{4}$ $\frac{1-2\text{a}}{4}$ $\frac{1+\text{a}}{4}$
→
If $\text{f}:\Big(-\frac{\pi}{2},\frac{\pi}{2}\Big)\rightarrow\text{R}$ and g $: [-1, 1] \rightarrow R$ be defined as f(x) = tanx and $\text{g(x)}=\sqrt{1-\text{x}^2}$ respectively, describe fog and gof.
→
A hospital dietician wishes to find the cheapest combination of two foods, $A$ and $B$, that contains at least $0.5$ milligram of thiamin and at least $600$ calories. Each unit of A contains $0.12$ milligram of thiamin and $100$ calories, while each unit of $B$ contains $0.10$ milligram of thiamin and $150$ calories. If each food costs $10$ paise per unit, how many units of each should be combined at a minimum cost?
→