f(x)=5^x\nf(x)=f(x)\n∫₀³f(x)dx=
f(x)=5^x\nf(x)=f(x)\n∫₀³f(x)dx=
Answer
Explanation:
Step1: Find the derivative of $F(x)$
The derivative of $a^x$ is $a^x\ln a$. So, if $F(x)=5^x$, then $f(x)=F'(x)=5^x\ln 5$.
Step2: Evaluate the definite - integral
We know that $\int_{0}^{3}f(x)dx=\int_{0}^{3}5^x\ln 5dx$. Since $f(x)$ is the derivative of $F(x) = 5^x$, by the fundamental theorem of calculus $\int_{a}^{b}F'(x)dx=F(b)-F(a)$. Here, $a = 0$, $b = 3$ and $F(x)=5^x$. So $\int_{0}^{3}5^x\ln 5dx=5^3 - 5^0$.
Step3: Calculate the result
$5^3=125$ and $5^0 = 1$. Then $5^3-5^0=125 - 1=124$.
Answer:
$124$