Integrals 4

Calculus Level 3

$\displaystyle \int_{-10}^{10} \frac {3^x}{ 3^{\lfloor x \rfloor} } \ \mathrm d x = \frac {a} { \ln 3 }$

What is the value of $a$ ?

Details and Assumptions

$\lfloor x \rfloor$ denote the floor function of $x$ .

1 solution

Jake Lai
Mar 17, 2015

A basic property of integrals is that you can split 'em up, so

$\int_{-10}^{10} \frac{3^{x}}{3^{\lfloor x \rfloor}} \ dx = \sum_{n=-10}^{9} \int_{n}^{n+1} \frac{3^{x}}{3^{\lfloor x \rfloor}} \ dx$

$= \sum_{n=-10}^{9} \frac{3^{n+1}}{3^{n}\ln 3}-\frac{3^{n}}{3^{n}\ln 3}$

$= 20 \left( \frac{3-1}{\ln 3} \right) = \frac{40}{\ln 3}$

after simplification. Thus, $a = \boxed{40}$ .

Jake Lai If you want to know why $\displaystyle \int a^{x} \ dx = \frac{a^{x}}{\ln a}+C$ , just use the FT of C in conjuction with chain rule.

$\frac{d}{dx} \ln a^{x} = \frac{\frac{d}{dx} a^{x}}{a^{x}}$

by the ever-lovely CR and

$\frac{d}{dx} \ln a^{x} = \frac{d}{dx} x\ln a = \ln a$

Equating the two,

$\ln a = \frac{\frac{d}{dx} a^{x}}{a^{x}} \longrightarrow \frac{d}{dx} a^{x} = a^{x}\ln a$

Jake Lai - 6 years, 2 months ago

"Jake Lai If you want to know why...", there's another you?

Pi Han Goh - 6 years, 2 months ago