Question

find the indefinite integral. check by differentiating.
k
∫23e^u du
∫23e^u du=□

Explanation:

Step1: Apply the integration formula

The integral formula for \(\int a e^{u}du\) (where \(a\) is a constant) is \(a e^{u}+C\). Here \(a = 23\), so \(\int 23e^{u}du=23\int e^{u}du\).
Since \(\int e^{u}du=e^{u}+C\), then \(\int 23e^{u}du = 23e^{u}+C\).

Step2: Check by differentiation

Differentiate \(y = 23e^{u}+C\) with respect to \(u\).
Using the rule \(\frac{d}{du}(e^{u})=e^{u}\) and the constant - multiple rule \(\frac{d}{du}(a\cdot f(u))=a\cdot\frac{d}{du}(f(u))\) (where \(a = 23\) and \(f(u)=e^{u}\)), we have \(\frac{dy}{du}=\frac{d}{du}(23e^{u}+C)\).
\(\frac{dy}{du}=23\frac{d}{du}(e^{u})+\frac{d}{du}(C)\).
Since \(\frac{d}{du}(C) = 0\) (where \(C\) is a constant) and \(\frac{d}{du}(e^{u})=e^{u}\), we get \(\frac{dy}{du}=23e^{u}\).

Answer:

\(23e^{u}+C\)