Question

find the indefinite integral. check by differentiating.
int 13 e ^ { u } d u
int 13 e ^ { u } d u =

Explanation:

Step1: Use the constant multiple rule

The constant multiple rule of integration states that \(\int kf(u)du = k\int f(u)du\) where \(k = 13\) and \(f(u)=e^{u}\). So, \(\int 13e^{u}du=13\int e^{u}du\).

Step2: Integrate \(e^{u}\)

We know that the integral of \(e^{u}\) with respect to \(u\) is \(e^{u}+C\). So, \(13\int e^{u}du = 13(e^{u}+C)\).
Since \(13C\) is also a constant, we can write the result as \(13e^{u}+C\).

Step3: Check by differentiating

Differentiate \(y = 13e^{u}+C\) with respect to \(u\). Using the sum rule \(\frac{d}{du}(f(u)+g(u))=\frac{d}{du}f(u)+\frac{d}{du}g(u)\) and the constant multiple rule \(\frac{d}{du}(kf(u))=k\frac{d}{du}f(u)\) (where \(k = 13\) and \(f(u)=e^{u}\), and \(\frac{d}{du}(C)=0\)).
We know that \(\frac{d}{du}(e^{u})=e^{u}\), so \(\frac{d}{du}(13e^{u}+C)=13\frac{d}{du}(e^{u})+\frac{d}{du}(C)=13e^{u}+0 = 13e^{u}\).

Answer:

\(\int 13e^{u}du=13e^{u}+C\)