What is area e^x,e^{-3x},ln(6) ?

The answer to area e^x,e^{-3x},ln(6) is 0.33758…

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area e^x,e^{-3x},ln(6)

area

e^{x}, e^{- 3 x}, ln (6)

0.33758 \dots

Solution steps

To find the intersection points solve

f_{i} (x) = f_{j} (x)

e^{x} = e^{- 3 x} : x = 0

e^{x} = ln (6) : x = ln (ln (6))

e^{- 3 x} = ln (6) : x = - \frac{ln ( ln ( 6 ) )}{3}

= \int_{- \frac{l n ( l n ( 6 ) )}{3}}^{0} ∣ f_{2} (x) - f_{3} (x) ∣ d x + \int_{0}^{l n (l n (6))} ∣ f_{1} (x) - f_{3} (x) ∣ d x

= \int_{- \frac{l n ( l n ( 6 ) )}{3}}^{0} e^{- 3 x} - ln (6) d x + \int_{0}^{l n (l n (6))} ∣ e^{x} - ln (6) ∣ d x

\int_{- \frac{l n ( l n ( 6 ) )}{3}}^{0} e^{- 3 x} - ln (6) d x = 0.08439 \dots

\int_{0}^{l n (l n (6))} ∣ e^{x} - ln (6) ∣ d x = 0.25319 \dots

= 0.08439 \dots + 0.25319 \dots

Simplify

= 0.33758 \dots

\frac{d}{d x} (\frac{x}{3 a} (a - x))

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f^{^{'}} (x) = x \cdot 1 5^{x}

\frac{x}{2 x + 3}

\int_{1}^{16} \frac{6}{x} d x

What is area e^x,e^{-3x},ln(6) ?
The answer to area e^x,e^{-3x},ln(6) is 0.33758…