_0^ /2 x x ,dx ^2 x + 3 x + 2 = - Vidyadip

MCQ

$\int_0^{\pi /2} {\frac{{\sin x\cos x\,dx}}{{{{\cos }^2}x + 3\cos x + 2}}} = $

A
$\log \left( {\frac{8}{9}} \right)$
✓
$\log \left( {\frac{9}{8}} \right)$
C
$\log (8 \times 9)$
D
None of these

✓

Answer

Correct option: B.

$\log \left( {\frac{9}{8}} \right)$

b
(b) Let $I = \int_0^{\pi /2} {\frac{{\sin x\cos x.dx}}{{{{\cos }^2}x + 3\cos x + 2}}} $

We put $\cos x = t \Rightarrow - \sin x\,dx = dt,$ then

$I = \int_0^1 {\frac{{t.dt}}{{{t^2} + 3t + 2}} = \int_0^1 {\left[ {\frac{2}{{t + 2}} - \frac{1}{{t + 1}}} \right]} } \,dt$

$ = [2\log (t + 2) - \log (\,t + 1)]_0^1$

$ = [2\log 3 - \log 2 - 2\log 2]$

$ = [2\log 3 - 3\log 2] = [\log 9 - \log 8] $

$= \log \left( {\frac{9}{8}} \right)$.

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

$\int {\frac{{{x^2}}}{{\left( {{x^2} + 1} \right)\left( {{x^2} + 4} \right)}}\,} dx$ is equal to

A coin is biased so that the head is $3$ times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of the coin, then the mean of $X$ is

Let $f$ be a function such that $f(x)\, = \,\sum\limits_{n\, - \,1}^n {\left[ {r\, + \,\cos \frac{x}{r}} \right]} $ where [.] denotes greatest integer function and $x \in [0,\pi]$, then range of $f(x)$ is-

Let $\vec{\alpha}=4 \hat{ i }+3 \hat{ j }+5 \hat{ k }$ and $\vec{\beta}=\hat{ i }+2 \hat{ j }-4 \hat{ k }$. Let $\vec{\beta}_1$ be parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ be perpendicular to $\vec{\alpha}$. If $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$, then the value of $5 \vec{\beta}_2 \cdot(\hat{ i }+\hat{ j }+\hat{ k })$ is

Let the foci and length of the latus rectum of an ellipse $\frac{\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}$ be $( \pm 5,0)$ and $\sqrt{50}$, respectively. Then, the square of the eccentricity of the hyperbola $\frac{\mathrm{x}^2}{\mathrm{~b}^2}-\frac{\mathrm{y}^2}{\mathrm{a}^2 \mathrm{~b}^2}=1$ equals

Let $f : R \rightarrow R$ and $g : R \rightarrow R$ be two functions defined by $f(x)=\log _{e}\left(x^{2}+1\right)-e^{-x}+1$ and $g(x)=\frac{1-2 e^{2 x}}{e^{x}}$. Then, for which of the following range of $\alpha$, the inequality $f\left(g\left(\frac{(\alpha-1)^{2}}{3}\right)\right)>f\left(g\left(\alpha-\frac{5}{3}\right)\right)$ holds?

If $\omega $ is a complex cube root of unity, then $(x - y)(x\omega - y)$ $(x{\omega ^2} - y) = $

If $A + B + C = \frac{\pi }{2}$ ,then value of $tanA\,\, tanB + tanB\,\, tanC + tanC\,\, tanA$ is

In a tournament there are twelve players $P_1, P_2, P_3,........ P_{12}$ and divided into six pairs at random. From each game a winner is decided on the basis of game played between the two players of the pair. Assuming each player is of equal strength, then the probability that exactly one out of $P_1$ and $P_2$ is among the losers is

If a circle of constant radius $3k$ passes through the origin and meets the axes at $A$ and $B$, the locus of the centroid of the triangle $OAB$ is the circle