Visit the website 123iitjee.manishverma.site for latest posts, courses, admission & more.

$\int {\frac{{\cos x - \cos 2x}}{{1 + 2\cos x}}dx}$=?

The given integral = $\int {\frac{{2\sin \frac{{3x}}{2}\sin \frac{x}{2}}}{{1 + 2 - 4{{\sin }^2}\frac{x}{2}}}dx}$=$\int {\frac{{2\sin \frac{{3x}}{2}\sin \frac{x}{2}}}{{3 - 4{{\sin }^2}\frac{x}{2}}}dx}$

=$\int {\frac{{2\sin \frac{{3x}}{2}\sin \frac{x}{2}.\sin \frac{x}{2}}}{{3\sin \frac{x}{2} - 4{{\sin }^3}\frac{x}{2}}}dx}$

=$\int {\frac{{2\sin \frac{{3x}}{2}{{\sin }^2}\frac{x}{2}}}{{\sin \frac{{3x}}{2}}}dx}$

=$\int {2{{\sin }^2}\frac{x}{2}dx}$

=$\int {(1 - \cos x)dx}$

=$x - \sin x + C$

Sum of the coefficients in the expansion of $(x+y)^n$ ....

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$