Chapter 4: Q27MP (page 239)

Show that $\frac{d}{d x} \int_{c o s x}^{s i n x} \sqrt{1 - t^{2}} d t = 1$ .

Short Answer

Expert verified

It is proved that $\frac{d}{d x} \int_{\cos x}^{\sin x} \sqrt{1 - t^{2}} d t = 1$ .

Step by step solution

Given Information

Given that . $\frac{d}{d x} \int_{\cos x}^{\sin x} \sqrt{1 - t^{2}} d t$ ...(1)

Formula Used

We know that $\frac{d}{d x} \int_{u (x)}^{v (x)} f (x, t) d t = f (x, v) \frac{d v}{d x} - f (x, u) \frac{d u}{d x} + \int_{u}^{v} \frac{\partial f}{\partial x} d t$ ...(2)

Solving the given equation

Compare equation (1) and equation (2).

\frac{d}{d x} \int_{\cos x}^{\sin x} \sqrt{1 - t^{2}} d t = \sqrt{1 - \sin^{2} x} . \frac{d}{d x} (\sin x) - \sqrt{1 - \cos^{2} x} . \frac{d}{d x} (\cos x) + \int_{\cos x}^{\sin x} 0 . d t \frac{d}{d x} \int_{\cos x}^{\sin x} \sqrt{1 - t^{2}} d t = \sqrt{\cos^{2} x} . (\cos x) - \sqrt{\sin^{2} x} . (- \sin x) \frac{d}{d x} \int_{\cos x}^{\sin x} \sqrt{1 - t^{2}} d t = (\cos x) . (\cos x) - (\sin x) . (- \sin x) \frac{d}{d x} \int_{\cos x}^{\sin x} \sqrt{1 - t^{2}} d t = \cos^{2} x + \sin^{2} x \frac{d}{d x} \int_{\cos x}^{\sin x} \sqrt{1 - t^{2}} d t = 1

Hence Proved

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Show that $\frac{d}{d x} \int_{c o s x}^{s i n x} \sqrt{1 - t^{2}} d t = 1$ .

Short Answer

Step by step solution

Given Information

Formula Used

Solving the given equation

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Short Answer

Step by step solution

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Formula Used

Solving the given equation

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Most popular questions from this chapter

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Show that $\frac{d}{d x} \int_{c o s x}^{s i n x} \sqrt{1 - t^{2}} d t = 1$ .