1. Tardigrade
  2. Question
  3. Mathematics
  4. For any integer n , the integral ∫ limits0πe cos2x cos3(2n+1)x dx has the value

Q. For any integer $ n $ , the integral $ \int\limits_{0}^{\pi}e^{\cos^2x}\cos^{3}\left(2n+1\right)x\, dx $ has the value

AMUAMU 2015Integrals Report Error

Solution:

Let $f(x)=e^{\cos ^{2} x} \cdot \cos ^{3}(2 n+1) x$
Then, $f(\pi-x)=e^{\cos ^{2}(\pi-x)} \cdot \cos ^{3}[(2 n+1) \pi-(2 n+1) x]$
$=-e^{\cos ^{2} x} \cdot \cos ^{3}(2 n+1) x$
$\Rightarrow f(\pi-x)=-f(x)$
Hence, $\int\limits_{0}^{\pi} e^{\cos ^{2} x} \cdot \cos ^{3}(2 n+1) x d x=0$
$\left[\because \int\limits_{0}^{2 a} f(x) d x=0\right.$, if $\left.f(2 a-x)=-f(x)\right]$

Questions from AMU 2015

1. For all real $ x, 4^{\sin^2x}+ 4\cos^{2x} $ is

2. The length of the perpendicular drawn from $(1,2,3)$ to the line $ \frac{x-6}{3} = \frac {y-7}{2} = \frac {z-7}{-2} $ is

Three Dimensional Geometry

3. The number of equivalence relations on the set $\{1,2,3\}$ containing $(1,2)$ and $(2,1)$ is

Relations and Functions - Part 2

4. Let $A=\left\{x \in R ; x \geq \frac{1}{2}\right\}$ and $B=\left\{x \in R ; x \geq \frac{3}{4}\right\}$. If $f: A \rightarrow B$ is defined as $f(x)=x^{2}-x+1$, then the solution set of the equation $f(x)=f^{-1}(x)$ is

Relations and Functions - Part 2

5. The complex number $ z = x + iy $ satisfy the equation $ |\frac {z-5i}{z+5i}| = 1 $ lies on

Complex Numbers and Quadratic Equations

Questions from Integrals

1. If $\Delta (x) \begin{vmatrix}1&\cos x&1 -\cos x\\ 1+ \sin x& \cos x &1+ \sin x - \cos x\\ \sin x &\sin x&1\end{vmatrix},$ then $ \int\limits^{\pi / 4}_{0} \Delta\left(x\right)dx $ is equal to

VITEEE 2014

2. Let $f \left(x\right)=x^{2}-2.$ if $^{6}\int_{3}f \left(x\right)dx=3f \left(c\right)$ for some $c \in (3, 6)$, then the value of $c$ is equal to

KEAM 2013

3. 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

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

WBJEE 2013

5. 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

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

KEAM 2016 Integrals

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

KEAM 2007 Integrals

Latest Updates