For any integer n, the integral _0^ e^ ^2 x ^3 (2n + 1)x ,dx = - Vidyadip

MCQ

For any integer $n,$ the integral $\int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x\,dx = } $

A
$ - 1$
✓
$0$
C
$1$
D
$\pi $

✓

Answer

Correct option: B.

$0$

b
(b) Let $f(x) = \int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x.dx} $

Since $\cos (2n + 1)(\pi - x) = \cos [(2n + 1)\pi - (2n + 1)x]$

$ = - \cos (2n + 1)x$ and ${\sin ^2}(\pi - x) = {\sin ^2}x$

Hence by the property of definite integral,

$\int_0^\pi {{e^{{{\sin }^2}x}}{{\cos }^3}(2n + 1)x\,dx = 0} $, $[f(2a - x) = - f(x)]$.

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 STD 12 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

Let $R = \{(1, 3), (2, 2), (3, 2)\}$ and $S = \{(2, 1), (3, 2), (2, 3)\}$ be two relations on set $A = \{1, 2, 3\}$. Then $RoS =$

If $f(x) = \frac{1}{{\sqrt {x + 2\sqrt {2x - 4} } }} + \frac{1}{{\sqrt {x - 2\sqrt {2x - 4} } }}$ for $x > 2$, then $f(11) = $

If $y = x + e^x$ , then at $x = 1$ , $\frac{{{d^2}x}}{{d{y^2}}}$ is equal to

If $(\text{a}+\text{b}-\text{x})=\text{f}(\text{x}),$ then $\int\limits^\text{b}_\text{a}\text{x f}(\text{x})\text{dx}$ is equal to :

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched

Interval Function

If the angle between the lines, $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $\frac{{5 - x}}{{ - 2}} = \frac{{7y - 14}}{p} = \frac{{z - 3}}{4}$ is ${\cos ^{ - 1}}\,\left( {\frac{2}{3}} \right),$ then $p$ is equal to

If $y = x^n,$ then the ratio of relative errors in $y$ and $x$ is :

The value of $\hat{\text{i}}.\big(\hat{\text{j}}\times\hat{\text{k}}\big)+\hat{\text{j}}.\big(\hat{\text{i}}\times\hat{\text{k}}\big)+\hat{\text{k}}.\big(\hat{\text{i}}\times\hat{\text{j}}\big),$ is:

The direction cosines of the line passing through $P(2, 3, -1)$ and the origin are:

The sum of lengths of the hypotenuse and another side of a right angled triangle is given. The area of the triangle will be maximum if the angle between them is