6 _0^|( 3 x+ 2 x+ x)| dx is equal to.......

6 _0^|2 2 x x+ 2 x| dx\\ 6 _0^|4 x ^2 x+2 x x| d x\\ I=12 _0^ x|2 ^2 x+ x|\\ Put x=t,- x dx=dt\\ $ aligned & I=-12 _1^-1|2 t^2+t| d t \\ & I= 12(_-1^-1 / 2(2 t^2+t) d t+_-1 / 2^0-(2 t^2+t) d t+_0^1(2 t^2+t) d t) aligned $ \\ I=17

6 int 0 pi sin 3 x sin 2 x sin x dx is equal to : null (Mathematics)

\quad 6 \int_{0}^{\pi}|2 \sin 2 \mathrm{x} \cos \mathrm{x}+\sin 2 \mathrm{x}| \mathrm{dx}\\

6 \int_{0}^{\pi}\left|4 \sin x \cos ^{2} x+2 \sin x \cos x\right| d x\\

I=12 \int_{0}^{\pi} \sin x\left|2 \cos ^{2} x+\cos x\right|\\

Put

\cos \mathrm{x}=\mathrm{t},-\sin \mathrm{x} \mathrm{dx}=\mathrm{dt}\\

\begin{aligned} & I=-12 \int_{1}^{-1}\left|2 t^{2}+t\right| d t \\ & I= 12\left(\int_{-1}^{-1 / 2}\left(2 t^2+t\right) d t+\int_{-1 / 2}^0-\left(2 t^2+t\right) d t+\int_0^1\left(2 t^2+t\right) d t\right) \end{aligned}

\\ \mathrm{I}=17

Explanation

The minimum number of elements that must be added to the relation R = {(a,b),(b,c),(b,d)} on the set { a,b,c,d} so that it is an equivalence relation is

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In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspaper is

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