If-f-x-determinant-sin-x-sin2x-sin-3x-sin-2x-sin-3x-3-4-sin-x-3-4-sin-x-1-sin-x-sin-x-1-then-the-value-of-0-pi-2-f-x-dx-is-

Question Number 19148 by khamizan833@yahoo.com last updated on 06/Aug/17

If

f (x) = | \begin{matrix} \sin x + \sin 2 x + \sin 3 x & \sin 2 x & \sin 3 x \\ 3 + 4 \sin x & 3 & 4 \sin x \\ 1 + \sin x & \sin x & 1 \end{matrix} |

then the value of \underset{0}{\int^{π / 2}} f (x) d x is

Answered by ajfour last updated on 06/Aug/17

f (x) = | \begin{matrix} \sin x + \sin 2 x + \sin 3 x & \sin 2 x & \sin 3 x \\ 3 + 4 \sin x & 3 & 4 \sin x \\ 1 + \sin x & \sin x & 1 \end{matrix} |

C_{1} \to C_{1} - C_{2} - C_{3}

f (x) = | \begin{matrix} \sin x & \sin 2 x & \sin 3 x \\ 0 & 3 & 4 \sin x \\ 0 & \sin x & 1 \end{matrix} |

\Rightarrow f (x) = \sin x (3 - 4 \sin^{2} x)

= \sin 3 x

\int_{0}^{π / 2} f (x) dx = \int_{0}^{π / 2} \sin 3 xdx

= - (\frac{\cos 3 x}{3}) ∣_{0}^{π / 2}

\int_{0}^{π / 2} f (x) dx = \frac{1}{3} .

Facebook Tweet Pin