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I'm having trouble understanding the derivation of the integral for Coefficient of Friction in Example 2.2 of John Anderson's Fundamentals of Aerodynamics 5e. I'm not sure how the author cancels $$ V_\infty^2c $$ and places the freestream velocity inside the integral. $$ \frac{u_2}{V_\infty}(\frac{u_1}{V_\infty}-\frac{u_2}{V_\infty}) $$ Equations below:

Solution
From Equation $(2.84)$ $$C_f=\frac{D'}{q_{\infty} c}=\frac{\rho_{\infty}}{\frac{1}{2}\rho_{\infty} V^2_{\infty}c}\int_0^{\delta}u_2(u_1-u_2)dy$$ where the integral is evaluated at the trailing edge of the plate. Hence, $$C_f=1\int_0^{\delta/2}\frac{u_2}{V_{\infty}}\left(\frac{u_1}{V_{\infty}}-\frac{u_2}{V_{\infty}}\right)d\left(\frac yc\right)$$

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  • $\begingroup$ Hello! It is preferable to type out screenshots; for formulae, one can use MathJax. I have edited your question accordingly. Thanks! $\endgroup$
    – jng224
    Commented May 28, 2021 at 18:20

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This is quite simple. Since $\dfrac{1}{V_\infty^2c}$ is a constant, it can be inserted into the integral. Thus, one $\dfrac{1}{V_\infty}$ goes into the factor $$\frac{u_2}{V_\infty},$$ the other $\dfrac{1}{V_\infty}$ goes into the factor $$\left(\frac{u_1}{V_\infty} - \frac{u_2}{V_\infty}\right),$$ and the $\dfrac{1}{c}$ goes into $$d\left(\frac{y}{c}\right).$$ Notice that the upper limit of the integral becomes $\delta/c$ and not $\delta/2$.

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  • $\begingroup$ Of course! Brilliant, thank you so much! @Tofi $\endgroup$
    – Tom
    Commented May 28, 2021 at 19:00
  • $\begingroup$ @Tom Glad to help 👍 $\endgroup$
    – Tofi
    Commented May 28, 2021 at 19:17

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