Download App

7.1 inderfinite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.1 inderfinite integral1 Mark
MCQ
$\int {\frac{{x\,\,dx}}{{{x^2} + 4x + 5}} = } $
  • A
    $\frac{1}{2}\log ({x^2} + 4x + 5) + 2{\tan ^{ - 1}}(x) + c$
  • B
    $\frac{1}{2}\log ({x^2} + 4x + 5) - {\tan ^{ - 1}}(x + 2) + c$
  • C
    $\frac{1}{2}\log ({x^2} + 4x + 5) + {\tan ^{ - 1}}(x + 2) + c$
  • ✓
    $\frac{1}{2}\log ({x^2} + 4x + 5) - 2{\tan ^{ - 1}}(x + 2) + c$
✓

Answer

Correct option: D.
$\frac{1}{2}\log ({x^2} + 4x + 5) - 2{\tan ^{ - 1}}(x + 2) + c$
d
(d)$I = \int {\frac{{x\,\,dx}}{{{x^2} + 4x + 5}}} $$ = \int {\frac{{x + 2 - 2\,\,}}{{{{(x + 2)}^2} + 1}}dx} $
$ = \frac{1}{2}\int {\frac{{2(x + 2)\,\,\,dx}}{{{{(x + 2)}^2} + 1}}} - 2\int {\frac{{dx}}{{1 + {{(x + 2)}^2}}}} $
$ = \frac{1}{2}\int {\frac{{dt}}{t} - 2\int {\frac{{dx}}{{1 + {{(x + 2)}^2}}}} } $

[Put $1 + {(x + 2)^2} = t$ in first expression ==> $2(x +2)dx = dt$]

$ = \frac{1}{2}\log t - 2{\tan ^{ - 1}}(x + 2) + c$
$ = \frac{1}{2}\log ({x^2} + 4x + 5) - 2{\tan ^{ - 1}}(x + 2) + 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 7.1 inderfinite integral→7.1 inderfinite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

$\int_{}^{} {\frac{{dx}}{{\sin x + \sqrt 3 \cos x}}} = $
→
For a steamer the consumption of petrol (per hour) varies as the cube of its speed (in km). If the speed of the current is steady at $C \,\,km/hr$ then the most economical speed of the steamer going against the current will be ........... $C$.
→
If $\text{f}(\text{x})=\tan^{-1}(\text{g}(\text{x})),$ where g(x) is monotonically increasing for $0<\text{x}<\frac{\pi}{2}.$ Then, f(x) is:
  1. Increasing on $\Big(0,\frac{\pi}{2}\Big)$
  2. Decreasing on $\Big(0,\frac{\pi}{2}\Big)$
  3. Increasing on $\Big(0,\frac{\pi}{4}\Big)$ and decreasing on $\Big(\frac{\pi}{4},\frac{\pi}{2}\Big)$
  4. None of these.
→
What is direction of vector $\vec{\text{a}}$ if it is multiplied with $-\lambda$:
  1. Downwards
  2. Upwards
  3. Same
  4. Opposite
→
Let $f: R \rightarrow R$ be a function defined by $f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)\left(2+x^{25}\right)\right)^{\frac{1}{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, the the greatest integer less than or equal to $g (1)$ is
→
Let for any three distinct consecutive terms $a, b, c$ of an $A.P,$ the lines $a x+b y+c=0$ be concurrent at the point $\mathrm{P}$ and $\mathrm{Q}(\alpha, \beta)$ be a point such that the system of equations $ x+y+z=6, $ $ 2 x+5 y+\alpha z=\beta$ and $x+2 y+3 z=4$, has infinitely many solutions. Then $(P Q)^2$ is equal to________.
→
$\Delta = \left| {\,\begin{array}{*{20}{c}}a&{a + b}&{a + b + c}\\{3a}&{4a + 3b}&{5a + 4b + 3c}\\{6a}&{9a + 6b}&{11a + 9b + 6c}\end{array}\,} \right|$where $a = i,b = \omega ,c = {\omega ^2}$, then $\Delta $is equal to
→
Consider the line $\mathrm{L}$ passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$ from the line $\mathrm{L}$ along the line $\frac{3 x-11}{2}=\frac{3 y-11}{1}=\frac{3 z-19}{2}$ is equal to :
→
Let  $a,b,c\; \in R.$ If $f\left( x \right) = a{x^2} + bx + c$ is such that $a + b + c = 3$ and $f\left( {x + y} \right) = f\left( x \right) + f\left( y \right) + xy,$ $\forall x,y \in R,$ then $\mathop \sum \limits_{n = 1}^{10} f\left( n \right)$ is equal to : 
→
Let $f(x)$ be a continuously differentiable function on the interval $(0, \infty)$ such that $f(1)=2$ and $\lim _{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1$

for each $x>0$. Then, for all $x>0, f(x)$ is equal to

→