Question

Find an antiderivative of f(x) = 3^2x + e^−3x whose graph passes through the
point. (0, 1/ln 9)

Answer 1

I will assume you mean f(x) = 3^(2x) + e^−(3x)

will 3^(2x) = 9^x
and the integral of 9^x is (1/ln9)(9^x)
the integral of e^(-3x) is (-1/3)e(-3x)

let g(x) be the antiderivative of f(x) = 3^2x + e^−3x
g(x) = (1/ln9)(9^x) - (1/3)e^(-3x) + c
1/ln 9 = (1/ln 9)(9^0) - (1/3) e^0 + c
1/ln 9 = (1/ln 9)(1) - (1/3) (1) + c
c = 1/3

g(x) = (1/ln9)(9^x) - (1/3)e^(-3x) + 1/3