1. Tardigrade
  2. Question
  3. Mathematics
  4. If b n =∫ limits0(π/2) ( cos 2 nx / sin x ) dx , n ∈ N, then

Q. If $b _{ n }=\int\limits_{0}^{\frac{\pi}{2}} \frac{\cos ^{2} nx }{\sin x } dx , n \in N$, then

JEE MainJEE Main 2022Integrals Report Error

Solution:

$b _{ n }=\int\limits_{0}^{\pi / 2} \frac{1+\cos 2 nx }{\sin x } dx$
$b _{ n +1}- b _{ n }=\int\limits_{0}^{\pi / 2} \frac{\cos ^{2}( n +1) x -\cos ^{2} nx }{\sin x } dx$
$=\int\limits_{0}^{\pi / 2} \frac{-\sin (2 n +1) x \sin x }{\sin x } dx$
$=\left(\frac{\cos (2 n +1) x }{2 n +1}\right)_{0}^{\pi / 2}=\frac{-1}{2 n +1}$
$\frac{1}{ b _{3}- b _{2}}, \frac{1}{ b _{4}- b _{3}}, \frac{1}{ b _{5}- b _{4}}$ are in A.P. with c.d. $=-2$

Questions from JEE Main 2022

1. Let the system of linear equations
$x+y+\alpha z=2$
$3 x+y+z=4 $
$x+2 z=1$
have a unique solution $\left(x^{*}, y^{*}, z^{*}\right)$. If $\left(\alpha, x^{*}\right),\left(y^{*}, \alpha\right)$ and $\left(x^{*},-y^{*}\right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is :

Determinants

2. Let $x, y > 0$. If $x^{3} y^{2}=2^{15}$, then the least value of $3 x +2 y$ is

Permutations and Combinations

3. Let $f(x)=\begin{cases} \frac{\sin(x - [x])}{x - [x]}, & x \in (-2, -1) \\\max \{2 x, 3[|x|]\} & |x| < 1 \\ 1 & \text { otherwise }\end{cases}$
where [ $t$ ] denotes greatest integer $\leq t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, then the ordered pair $( m , n )$ is :

Continuity and Differentiability

4. The value of the integral $\int\limits_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)}$ is equal to

Integrals

5. $\lim \limits_{n \rightarrow \infty}\left(\frac{n^{2}}{\left(n^{2}+1\right)(n+1)}+\frac{n^{2}}{\left(n^{2}+4\right)(n+2)}+\frac{n^{2}}{\left(n^{2}+9\right)(n+3)}+\ldots+\frac{n^{2}}{\left(n^{2}+n^{2}\right)(n+n)}\right)$ is equal to

Integrals

Questions from Integrals

1. The sum of the series $\frac{1}{1\times2} ^{^{25}}C_{0} + \frac{1}{2\times3}^{^{25}}C_{1} + \frac{1}{3\times4} ^{^{25}}C_{2} + ......... + \frac{1}{26\times27} ^{^{25}}C_{25}$

WBJEE 2013

2. The value of the integral
$ \int\limits^2_{1}e^{x} \left(log_{e} \,x + \frac{x+1}{x}\right)dx$ is

WBJEE 2013

3. If $ f(x)=\int_{1}^{x}{\sqrt{4-{{t}^{2}}}}\,\,\,dt, $ then real roots of the equation $ x-f'(x)=0 $ are

J & K CET 2012

4. If $f\left(x\right) = \int\limits^{sin\,x}_{2x}cos\left(t^{3}\right)dt$, then $f'{x}$ is equal to

KEAM 2012

5. $\int\limits_{0}^{1} x e^{-5x} \, dx$ is equal to

KEAM 2011

Mathematics Most Viewed Questions

1. The solution of $\frac{dy}{dx} = \frac{y}{x}+\tan \frac{y}{x}$ is

WBJEE 2011 Differential Equations

2. The solution of the differential equation $\frac{dy}{dx} = (x +y)^2$ is

COMEDK 2009 Differential Equations

3. If a and b are vectors such that $|a+b|=|a-b|$ then the angle between a and b is

KCET 2007 Vector Algebra

4. If $\begin{bmatrix}1&- \tan\theta \\ \tan \theta&1\end{bmatrix}\begin{bmatrix}1&\tan \theta \\ - \tan \theta &1\end{bmatrix}^{-1} = \begin{bmatrix}a&-b\\ b&a\end{bmatrix}$ then

COMEDK 2009 Matrices

5. $\int\frac{1}{\sin x\, \cos x}$ dx is equal to

KEAM 2016 Integrals

6. $\int\frac{\sin \frac{5x}{2}}{\sin \frac{x}{2}} dx $ is equal to :
(where $c$ is a constant of integration)

JEE Main 2019 Integrals

7. A particle is dropped under gravity from rest from a height $ h(g=9.8\,m/{{s}^{2}}) $ and it travels a distance $ \frac{9h}{25} $ in the last second the height $ h $ is:

Bihar CECE 2006

8. The value of $ \int{\frac{{{x}^{2}}+1}{{{x}^{4}}-{{x}^{2}}+1}}dx $ is

KEAM 2007 Integrals

9. The value of $\int \frac{x^2+1}{x^2-1}dx$ is

KCET 2007 Integrals

10. The value of the integral $\int\frac{cos x}{sin x + cos x}dx$ is equal to

KEAM 2013 Integrals

Latest Updates