First slide

Evaluation of definite integrals

Question

If $f (x) =$ $|\begin{array}{ccc} \sin x + \sin 2 x + \sin 3 x & \sin 2 x & \sin 3 x \\ 3 + 4 \sin x & 3 & 4 \sin x \\ 1 + \sin x & \sin x & 1 \end{array}|$ then the value of $\int_{0}^{π / 2} f (x) d x$ is

Moderate

A

3
B

0
C

2/3
D

1/3

Solution

By applying the operation $C_{1} \to C_{1} - C_{2} - C_{3}, f (x)$ can be written as
$\begin{array}{l} f (x) = |\begin{array}{ccc} \sin x & \sin 2 x & \sin 3 x \\ 0 & 3 & 4 \sin x \\ 0 & \sin x & 1 \end{array}| \\ = \sin x (3 - 4 \sin^{2} x) = \sin 3 x \end{array}$
So $\int_{0}^{π / 2} f (x) d x = - {\frac{\cos 3 x}{3}|}_{0}^{π / 2} = \frac{1}{3} .$

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