Answer
$$\int 5^t\sin(5^t)dt=-\frac{\cos(5^t)}{\ln5}+C$$
Work Step by Step
$$A=\int 5^t\sin(5^t)dt$$
Let $u=5^t$.
We would have $du=(5^t)'dt=5^t\ln 5dt$. Therefore, $5^tdt=\frac{1}{\ln 5}du$
Substitute into $A$, we have $$A=\frac{1}{\ln5}\int\sin udu$$ $$A=\frac{1}{\ln5}(-\cos u)+C$$ $$A=-\frac{\cos(5^t)}{\ln5}+C$$
