Chapter 1: Q12-1P (page 1)

If $y = \int_{0}^{x} s i n t^{2} d t$ , find $\frac{d y}{d x}$ .

Short Answer

Expert verified

The value of $\frac{d y}{d x}$ is $\frac{\sin x}{2 \sqrt{x}}$ .

Step by step solution

Given Information

Given that $y = \int_{0}^{\sqrt{x}} \sin t^{2} d t$ . ...(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)

Find dydx

Compare equation (1) and equation (2).

$f = \sin (t^{2}), u = 0 and v = \sqrt{x}$

Substitute values of f, u and v in the equation (2).

$\frac{d y}{d x} = f (\sqrt{x}) \frac{d \sqrt{x}}{d t} - f (0) \frac{d (0)}{d x} = \sin x \times \frac{1}{2} x^{- \frac{1}{2}} = \frac{\sin x}{2 \sqrt{x}}$

Hence the value of $\frac{d y}{d x}$ is $\frac{\sin x}{2 \sqrt{x}}$ .

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If $y = \int_{0}^{x} s i n t^{2} d t$ , find $\frac{d y}{d x}$ .

Short Answer

Step by step solution

Given Information

Formula Used

Find dydx

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If y=∫0xsint2dt, find dydx.

Short Answer

Step by step solution

Given Information

Formula Used

Find dydx

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Radiation

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If $y = \int_{0}^{x} s i n t^{2} d t$ , find $\frac{d y}{d x}$ .