If f(x)=| sin 2 x(1+2 cos x) sin 2 x sin 3 x 3+4 sin x 3 4 sin x 1

step 1 In this problem, we perform the following transformation on the determinant. We take column 1 as

If f(x)=| sin 2 x(1+2 cos x) sin 2 x sin 3 x ...

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

Key Concepts

Matrix Simplification Techniques

Matrix simplification techniques involve applying operations like row or column manipulation to transform a determinant into a simpler equivalent form. Recognizing patterns or cancellations within a matrix, especially when the entries are trigonometric functions, is crucial to reducing the complexity of the problem and making subsequent integration more manageable.

Definite Integration of Trigonometric Functions

Definite integration involves computing the accumulation of a function over an interval, and when trigonometric functions are involved, knowledge of their periodicity and symmetry can be exploited. This is particularly useful when the integrand results from a determinant of trigonometric expressions, enabling the application of standard integration techniques over a bounded interval.

Determinant Properties

Determinants are scalar quantities computed from square matrices that carry important algebraic and geometric information, such as invertibility and volume scaling. In problems where the function is defined by a determinant, properties like multilinearity and the effects of elementary row and column operations are essential for simplification and evaluation.

Trigonometric Identities

Trigonometric identities simplify expressions involving sine and cosine functions by relating multiple angles to single angles or reducing complex expressions into simpler ones. Using identities such as those for sin 2x or sin 3x often transforms the integrand into a form that is easier to work with both for algebraic manipulation and integration.

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