Which of the following statements is incorrect for the function $\mathrm{g}(\alpha)$ for $\alpha \in \mathrm{R}$ such that $g(\alpha)=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin ^{\alpha} x}{\cos ^{\alpha} x+\sin ^{\alpha} x} d x$

(C) $g(\alpha)$ is a strictly increasing function

(A) $\mathrm{g}(\alpha)$ is an even function

(B) $\mathrm{g}(\alpha)$ has an inflection point at $\alpha=-\frac{1}{2}$

(D) $\mathrm{g}(\alpha)$ is a strictly decreasing function

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Definite Integrals

Which of the following statements is incorrect for the function

g (α)

for

α \in R

such that

g (α) = \int_{\frac{π}{6}}^{\frac{π}{3}} \frac{s i n ^{α} x}{c o s ^{α} x + s i n ^{α} x} d x

Answer: (C)

Using the property

\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x

, we have

g (α) = \int_{π /6}^{π /3} \frac{s i n ^{α} x}{c o s ^{α} x + s i n ^{α} x} d x

. Let

x \to \frac{π}{2} - x

g (α) = \int_{π /6}^{π /3} \frac{c o s ^{α} x}{c o s ^{α} x + s i n ^{α} x} d x

. Adding:

2 g (α) = \int_{π /6}^{π /3} d x = \frac{π}{6}

, so

g (α) = \frac{π}{12}

is constant for all

α

. Therefore it is neither strictly increasing nor strictly decreasing. Statement (C) is incorrect.

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Single correct option