x , ,dx x^2 + 4x + 5 = - Vidyadip

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The equations of two sides of a variable triangle are $x =0$ and $y =3$, and its third side is a tangent to the parabola $y^2=6 x$. The locus of its circumcentre is :

Suppose $\quad f : R \rightarrow(0, \infty)$ be a differentiable function such that $5 f ( x + y )= f ( x ) \cdot f ( y ), \forall x , y \in R$. If $f(3)=320$, then $\sum \limits_{n=0}^5 f(n)$ is equal to :

The area of the region $S=\left\{(x, y): 3 x^{2} \leq 4 y \leq 6 x+24\right\} \text { is }...... \,.$

If $(a, a^2) $ falls inside the angle made by the lines $y\; = \frac{x}{2},\;x > \;0$ and $y\; = \;3x,\;x\; > \;0$ then $a$ belong to

All the points $(x, y)$ in the plane satisfying the equation $x^2+2 x \sin (x y)+1=0$ lie on

Solution of differential equation $\frac{{dy}}{{dx}} + ay = {e^{mx}}$ is

The number of terms which are free from radical signs in the expansion of ${({y^{1/5}} + {x^{1/10}})^{55}}$ is

The first term of an $A.P. $ is $2$ and common difference is $4$. The sum of its $40$ terms will be

The value of $\mathop {\lim }\limits_{x \to \infty } {x^{\frac{1}{3}}}\left( {{{\left( {x + 1} \right)}^{\frac{2}{3}}} - {{\left( {x - 1} \right)}^{\frac{2}{3}}}} \right)$ is

The ${n^{th}}$ term of series $\frac{1}{1} + \frac{{1 + 2}}{2} + \frac{{1 + 2 + 3}}{3} + .......$ will be