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Chapter 5: Q. 3 (page 463)

Show that if $x = \tan 鈦� u$ , then $d x = \sec^{2} 鈦� u d u$ , in the following two ways: (a) by using implicit differentiation, thinking of $u$ as a function of $x$ , and (b) by thinking of $x$ as a function of $u$ .

Short Answer

Expert verified

Ans: It is proved that $x = \tan 鈦� u, then d x = \sec^{2} 鈦� u d u$ by (a) and (b) both ways.

Step by step solution

01

Step 1. given information.

given,

$x = \tan 鈦� u, then d x = \sec^{2} 鈦� u d u$

02

Step 2. (a) The objective is to show that dx=sec2⁡udu by using implicit differentiation by assuming u as a function of x.

The differentiation is done by assuming $u$ as a function of $x$ .

The differentiation is shown below.

$\begin{aligned} x = \tan 鈦� u \\ 1 = \sec^{2} 鈦� u \frac{d u}{d x} \\ d x = \sec^{2} 鈦� u d u \end{aligned}$

Hence the expression is proved.

03

Step 3. (b) The objective is to show that dx=sec2⁡uduby using implicit differentiation by assuming x as a function of u.

The differentiation is done by assuming $x$ as a function of $u$ .

The differentiation is shown below.

$\begin{aligned} x = \tan 鈦� u \\ \frac{d x}{d u} = \sec^{2} 鈦� u \\ d x = \sec^{2} 鈦� u d u \end{aligned}$

Hence the expression is proved.

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Most popular questions from this chapter

Solve $鈭� \frac{1}{x^{2} + 9} d x$ the following two ways:

(a) with the trigonometric substitution x = 3 tan u;

(b) with algebra and the derivative of the arctangent.

Solve $鈭� \frac{x}{4 + x^{2}} d x$ the following two ways:

(a) with the substitution $u = 4 + x^{2};$

(b) with the trigonometric substitution x = 2 tan u.

Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.

Solve given definite integral.

${鈭�}_{1}^{2} 鈥� \frac{1}{x \sqrt{9 鈭� x^{2}}} d x$

Explain why, if $x = a s e c u$ , then $\sqrt{a^{2} s e c^{2} u 鈭� a^{2}}$ is $鈭� a \tan u$ if $x < 鈭� a$ and is $a \tan u$ if $x > a$ . Your explanation should include a discussion of domains and absolute values.

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