Question

find the indefinite integral. check by differentiating.
int 12 e ^ { u } d u
int 12 e ^ { u } d u = square

Explanation:

Step1: Use integral constant multiple rule

The integral constant multiple rule is $\int kf(u)du = k\int f(u)du$, where $k = 12$ and $f(u)=e^{u}$. So, $\int 12e^{u}du=12\int e^{u}du$.

Step2: Use integral formula for $e^{u}$

The integral formula is $\int e^{u}du=e^{u}+C$. Then $12\int e^{u}du = 12(e^{u}+C)=12e^{u}+C$ (since $12C$ is also a constant).

Step3: Check by differentiating

Differentiate $y = 12e^{u}+C$ using the rule $\frac{d}{du}(e^{u})=e^{u}$ and the constant multiple rule $\frac{d}{du}(kf(u))=k\frac{d}{du}(f(u))$. So, $\frac{d}{du}(12e^{u}+C)=12\frac{d}{du}(e^{u})+\frac{d}{du}(C)=12e^{u}+0 = 12e^{u}$.

Answer:

$12e^{u}+C$