If then equals

Engineering

Mathematics

Introduction to Determinants

Question

If $f (x) = ∣ \begin{array}{ccc} \sin x & 1 & 0 \\ 1 & 2 \sin x & 1 \\ 0 & 1 & 2 \sin x \end{array} ∣$ then $\int_{- \frac{π}{2}}^{\frac{π}{2}} f (x)$ equals

– 1

$\frac{3 π}{2}$

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$f (x) = ∣ \begin{array}{ccc} \sin x & 1 & 0 \\ 1 & 2 \sin x & 1 \\ 0 & 1 & 2 \sin x \end{array} ∣$
Expanding along first column we get,
f(x) = sinx(4sin²x – 1) – 1(2sinx – 0) = 4sin³x – 3sinx
Clearly f(– x) = f(x) ⇒ f is an odd function
Hence $\int_{- \frac{π}{2}}^{\frac{π}{2}} f (x)$

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