logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
0 votes

$I= \int \limits_0^{\frac{\pi}{6}} \cos ^5 3 \theta . \sin ^3 6 \theta d \theta$ Find the value of I.

\[\begin {array} {1 1} (a)\;\frac{16}{297} \\ (b)\;\frac{17}{297} \\ (c)\;\frac{1}{7} \\ (d)\;\frac{2}{11} \end {array}\]
Can you answer this question?
 
 

1 Answer

0 votes
$I= \int \limits_0^{\frac{\pi}{6}} \cos ^5 3 \theta . \sin ^3 6 \theta d \theta$
$3 \theta=t$
$ 3 d \theta=dt$
$I= \large\frac{1}{3} $$ \int \limits_0^{\frac{\pi}{2}} \cos ^5 t . \sin ^3 2t dt$
$\quad= \large\frac{1}{3} $$ \int \limits_0^{\frac{\pi}{2}} \cos ^5 t .(2 \sin t \cos t)^3 dt$
$\quad=\large\frac{8}{3} $$ \int \limits_0^{\frac{\pi}{2}} \cos ^8 t . \sin ^3 t dt$
$\cos t =x$
$ -\sin t dt=dx$
$\quad=\large\frac{8}{3} - $$ \int \limits_1^0 x^3(1-x^2)dx$
$\quad=\large\frac{8}{3} - $$ \int \limits_1^0 (x^3-x^{10})dx$
$\quad=\large\frac{8}{3}\bigg( \frac{1}{9} - \frac{1}{11}\bigg)$
$\quad=\large\frac{8}{3} \times \frac{2}{99}$
$\quad= \large\frac{16}{297}$
Hence a is the correct answer.
answered Dec 20, 2013 by meena.p
 
Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...