If f(x)= | beginmatrix sin x+ sin 2x+ sin 3x sin 2x sin 3x 3+4 sin x 3 4 sin x 1+ sin x sin x 1 endmatrix | , then the value of ∫0π /2f(x)dx is

Question

If f(x)= | beginmatrix sin x+ sin 2x+ sin 3x sin 2x sin 3x 3+4 sin x 3 4 sin x 1+ sin x sin x 1 endmatrix | , then the value of ∫0π /2f(x)dx is (A)  3  (B)  2/3  (C)  1/3  (D)  0

Tardigrade Admin · Accepted Answer

We have, f(x)=| beginmatrix sin x+ sin 2x+ sin 3x sin 2x sin 3x 3+4 sin x 3 4 sin x 1+ sin x sin x 1 endmatrix | Applying C1→ C1-C2-C3 , we get f(x)=| beginmatrix sin x sin 2x sin 3x 0 3 4 sin x 0 sin x 1 endmatrix | =3 sin x-4 sin 3x= sin 3x ∴ ∫0π /2f(x) dx=∫0π /2 sin 3x dx =[ -( cos 3x/3) ]0π /2 =-(1/3)[ cos (3π /2)- cos 0 ]=(1/3)

Q. If $f (x) =$ $∣ ∣ sin x + sin 2 x + sin 3 x 3 + 4 sin x 1 + sin x sin 2 x 3 sin x sin 3 x 4 sin x 1 ∣ ∣$ , then the value of $\int_{0}^{π /2} f (x) d x$ is

$3$

$2/3$

$1/3$

$0$

Q. If f(x)= ∣∣​sinx+sin2x+sin3x3+4sinx1+sinx​sin2x3sinx​sin3x4sinx1​∣∣​ , then the value of ∫0π/2​f(x)dx is

3

2/3

1/3

0

Q. If $f (x) =$ $∣ ∣ sin x + sin 2 x + sin 3 x 3 + 4 sin x 1 + sin x sin 2 x 3 sin x sin 3 x 4 sin x 1 ∣ ∣$ , then the value of $\int_{0}^{π /2} f (x) d x$ is

$3$

$2/3$

$1/3$

$0$