Question

In Exercises \(13-16,\) verify that \(\int f(u) d u \neq \int f(u) d x\) $$f(u)=e^{u}\( and \)u=7 x$$

Short answer

The integrals are \(\int e^{u} du = e^{u} + C\) and \(\int e^{7x} dx = \frac{1}{7}e^{7x} + C\), so indeed, \(\int f(u) du \neq \int f(u) dx\).

Step-by-step solution

Use Chain Rule to differentiate

Chain rule states that the derivative of \(f(g(x))\) where \(g(x)\) is a function, is \(f'(g(x)) \cdot g'(x)\). Given \(f(u) = e^{u}\) and \(u = 7x\), use Chain Rule to differentiate \(f(u)\) by \(x\). This gives \(f'(x) = \frac{d}{dx}(e^{7x}) = 7e^{7x}\).

Evaluate integral of \(f(u)\) with respect to \(u\)

Evaluate \(\int e^{u} du\). This integral evaluates to \(e^{u} + C\), where \(C\) is the constant of integration.

Evaluate integral of \(f(u)\) with respect to \(x\)

Evaluate \(\int e^{7x} dx = \frac{1}{7}e^{7x} + C\). Here, the integral was computed using the formula \(\int e^{ax}dx = \frac{1}{a}e^{ax} + C\).

Compare both integrals

Upon comparing the integrals, it can be noticed that \(\int e^{u} du \neq \int e^{7x} dx\). Thus, the statement \(\int f(u) du \neq \int f(u) dx\) is verified.