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Chapter 6: Q. 8 (page 569)

Explain, using the chain rule and / or u-substitution, why

鈭�1qydydxdx=鈭�1qydy.

Short Answer

Expert verified

The required result is鈭�1q(y)dydxdx=鈭�1q(y)dy

Step by step solution

01

Step 1. Given information 

The expression is鈭�1qydydxdx=鈭�1qydy

02

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

du=d[y(x)]=y'(x)dx=dydxdx

Replace this in the integral to obtain

鈭�1q(y)dydxdx=鈭�1q(u)du鈥�鈥�(2)

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

鈭�f(x)dx=鈭�f(t)dt

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

鈭�1q(u)du=鈭�1q(y)dy鈥�..(3)

Therefore, from (2) and (3)

鈭�1q(y)dydxdx=鈭�1q(y)dy

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