Chapter 5 - Section 5.4 - Indefinite Integrals and the Net Change Theorem - 5.4 Exercises - Page 409: 17

$$\int2^t(1+5^t)dt=\frac{2^t}{\ln2}+\frac{10^t}{\ln10}+C$$

$$A=\int2^t(1+5^t)dt$$ $$A=\int(2^t+2^t\times5^t)dt$$ $$A=\int(2^t+10^t)dt$$ From Table 1, $$\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx$$ Therefore, $$A=\int(2^t)dt+\int(10^t)dt$$ From Table 1, we also get the followings $$\int(b^x)dx=\frac{b^x}{\ln b}+C$$ Therefore, $$A=\frac{2^t}{\ln2}+\frac{10^t}{\ln10}+C$$