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Chapter 1: Q12-1P (page 1)

If $y = {鈭�}_{0}^{x} s i n t^{2} d t$ , 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

01

Given Information

Given that $y = {鈭�}_{0}^{\sqrt{x}} \sin t^{2} d t$ . ...(1)

02

Formula Used

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

03

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).

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

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

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Most popular questions from this chapter

Write the Maclaurin series for $\frac{1}{\sqrt{1 + x}}$ in $鈭�$ form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms ofn as we did in Example $2$ above

$f (x) = e^{x}, a = 3 .$

$\frac{1 + x}{1 - x}$

Prove theorem $14.3$ . Hint: Group the terms in the error as role="math" localid="1657423688910" $(a_{n + 1} + a_{n + 2}) + (a_{n + 3} + a_{n + 4}) + 鈰�$ to show that the error has the same sign as role="math" localid="1657423950271" $a_{n + 1} .$ Then group them asrole="math" localid="1657423791335" $a_{n + 1 + (a_{n + 2} + a_{n + 3}) + (a_{n + 4} + a_{n + 5}) + 鈰�}$ to show that the error has magnitude less than $| a_{n + 1} | .$

Use the ratio test to show that a binomial series converges for $| x | < 1$

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