Question

\(y = \int_{1/8}^{\sin^2 x} \sin^{-1}\sqrt{t} dt + \int_{1/8}^{\cos^2 x} \cos^{-1}\sqrt{t} dt, 0 \leq x \leq \pi/2\)

• A. Is the equation of a straight line parallel to the \(x\)-axis
• B. Is the equation of a straight line which is the bisector of first quadrant
• C. Is the equation of a straight line which is the bisector of second quadrant
• D. None of the above

Hint:

Use Leibniz rule and simplify using the given interval.

Answer 1

The Correct Option is A

To determine if the given expression is the equation of a straight line, we first need to analyze the expression:

\[ y = \int_{1/8}^{\sin^2 x} \sin^{-1}\sqrt{t} \, dt + \int_{1/8}^{\cos^2 x} \cos^{-1}\sqrt{t} \, dt \]

We will differentiate \( y \) with respect to \( x \) to investigate its dependency on \( x \).

1. The derivative of an integral of the form \( \int_a^{b(x)} f(t) \, dt \) with respect to \( x \) is given by the Leibniz rule:

\[ \frac{dy}{dx} = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x) \]

2. Applying this to the first integral \(\int_{1/8}^{\sin^2 x} \sin^{-1}\sqrt{t} \, dt\):

\[ \frac{d}{dx} \left( \int_{1/8}^{\sin^2 x} \sin^{-1}\sqrt{t} \, dt \right) = \sin^{-1} \sqrt{\sin^2 x} \cdot \frac{d}{dx}(\sin^2 x) \]

Simplifying, we get:

\[ \sin^{-1} |\sin x| \cdot 2\sin x \cos x = \sin^{-1}(\sin x) \cdot 2\sin x \cos x = x \cdot \sin(2x) \]

3. Now, consider the second integral \(\int_{1/8}^{\cos^2 x} \cos^{-1}\sqrt{t} \, dt\):

\[ \frac{d}{dx} \left( \int_{1/8}^{\cos^2 x} \cos^{-1}\sqrt{t} \, dt \right) = \cos^{-1} \sqrt{\cos^2 x} \cdot \frac{d}{dx}(\cos^2 x) \]

Simplifying, we have:

\[ \cos^{-1} |\cos x| \cdot (-2\sin x \cos x) = \cos^{-1}(\cos x) \cdot (-2\sin x \cos x) = ( \dfrac{\pi}{2} - x ) \cdot (-\sin(2x)) \]

Combining both derivatives:

\[ \frac{dy}{dx} = x \cdot \sin(2x) + \left( \dfrac{\pi}{2} - x \right) \cdot (-\sin(2x)) \]

On simplifying:

\[ \frac{dy}{dx} = x \cdot \sin(2x) - \left(\dfrac{\pi}{2} - x\right) \cdot \sin(2x) \]

\[ \frac{dy}{dx} = \left(x - \left(\dfrac{\pi}{2} - x\right)\right)\cdot \sin(2x) = \left(2x - \dfrac{\pi}{2}\right)\cdot \sin(2x) \]

4. Since \(\frac{dy}{dx} = 0\), \(y\) is a constant.
5. This implies that the function \( y \) is independent of \( x \), and therefore represents a straight line parallel to the \( x \)-axis.
6. The correct answer is that the expression is the equation of a straight line parallel to the \( x \)-axis.