Chapter 6: Q. 8 (page 569)

Explain, using the chain rule and / or $u$ -substitution, why
$\int \frac{1}{q (y)} \frac{d y}{d x} d x = \int \frac{1}{q (y)} d y$ .

Short Answer

Expert verified

The required result is $\int \frac{1}{q (y)} \frac{d y}{d x} d x = \int \frac{1}{q (y)} d y$

Step by step solution

Step 1. Given information

The expression is $\int \frac{1}{q (y)} \frac{d y}{d x} d x = \int \frac{1}{q (y)} d y$

Step 2. Calculation

Utilize the substitution $u = y (x)$ and think about the integral on the left side of equation (1). Afterward, using the chain rule of distinction

$\begin{array}{rcl} d u & = & d [y (x)] \\ = & y^{'} (x) d x \\ = & \frac{d y}{d x} d x \end{array}$

Replace this in the integral to obtain

$\int \frac{1}{q (y)} \frac{d y}{d x} d x = \int \frac{1}{q (u)} d u \dots \dots (2)$

Keep in mind that integration is independent of the integration variable, therefore

$\int f (x) d x = \int f (t) d t$

Integrate the preceding result into the right side of equation (2) to obtain

$\int \frac{1}{q (u)} d u = \int \frac{1}{q (y)} d y \dots . . (3)$

Therefore, from (2) and (3)

$\int \frac{1}{q (y)} \frac{d y}{d x} d x = \int \frac{1}{q (y)} d y$

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Explain, using the chain rule and / or $u$ -substitution, why
$\int \frac{1}{q (y)} \frac{d y}{d x} d x = \int \frac{1}{q (y)} d y$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

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Explain, using the chain rule and / or u-substitution, why ∫1qydydxdx=∫1qydy.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

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Most popular questions from this chapter

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Explain, using the chain rule and / or $u$ -substitution, why
$\int \frac{1}{q (y)} \frac{d y}{d x} d x = \int \frac{1}{q (y)} d y$ .