Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS1 Mark
MCQ
Choose the correct option from given four options:
$\int\frac{\text{x}}{\text{x}+1}$ is equal to:
  • A
    $\text{x}+\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
  • B
    $\text{x}+\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
  • C
    $\text{x}-\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$
  • ✓
    $\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$
✓

Answer

Correct option: D.
$\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$
Let $\text{I}=\int\frac{\text{x}}{\text{x}+1}\text{dx}$

We know that, $\frac{\text{x}^3}{\text{x}+1}$ is an improper fraction.

To convert it into proper fraction, we have to divide numerator by denominator.

After performing long division, we get

$\frac{\text{x}^3}{\text{x}+1}=(\text{x}^2-\text{x}+1)-\frac{1}{(\text{x}+1)}$

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

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

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 "M.C.Q (1 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

The function $\text{f}:[0,\infty)\rightarrow\ \text{R}$ given by $\text{f(x)}=\frac{\text{x}}{\text{x}+1}$ is:
→
If $x\frac{{dy}}{{dx}} + y = x\frac{{f\left( {xy} \right)}}{{f'\left( {xy} \right)}}$ , then $f(xy)$ is equal to
→
Let $\vec{a}=\hat{i}+\alpha \hat{j}+3 \hat{k}$ and $\vec{b}=3 \hat{i}-\alpha \hat{j}+\hat{k} \cdot$ If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $8 \sqrt{3}$ square units, then $\overrightarrow{ a } \cdot \overrightarrow{ b }$ is equal to ....... .
→
A function $f$ satisfying $f ‘ \,(sin\, x)$ $= cos^2 x$ for all $x$ and $f(1) = 1$ is :
→
Consider the binary operation * defined on Q − {1} by the rule a * b = a + b − ab for all a, b ∈ Q − {1}. The identity element in Q − {1} is:
→
A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is
→
If $\text{A}=\frac{1}{3}\begin{bmatrix} 1 & 1 & 2 \\ 2 & 1 & -2 \\ \text{x} & 2 & \text{y} \end{bmatrix}$ is prthogonal, than x + y =
→
A and B are two events such that P(A) = 0.25 and P(B) = 0.50. The probability pf both happening together is 0.14. The probability of both A and B hot happening is.
→
Let $A$ and $E$ be any two events with positive probabilities:
Statement $- 1$: $P\left( {E/A} \right) \geq P\left( {A/E} \right)P\left( E \right)$
Statement $-2$ : $P\left( {A/E} \right) \geq P\left( {A \cap E} \right)$
→
Lef $f:(0, \pi) \rightarrow R$ be a function given by

$f(x)=\left\{\begin{array}{cc}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}, & 0 < x < \frac{\pi}{2} \\ a-8, & x=\frac{\pi}{2} \\ (1+\mid \cot x)^{\frac{b}{a}|\tan x|}, & \frac{\pi}{2} < x < \pi\end{array}\right.$

Where $a, b \in Z$. If $f$ is continuous at $x=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to ..........

→