What is the integral from 3 to 4 of f(5^x)5^x ?

The integral from 3 to 4 of f(5^x)5^x is (187500f)/(ln(5))

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integral from 3 to 4 of f(5^x)5^x

\int_{3}^{4} f (5^{x}) 5^{x} d x

\frac{187500 f}{ln ( 5 )}

Solution steps

\int_{3}^{4} f \cdot 5^{x} \cdot 5^{x} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= f \cdot \int_{3}^{4} 5^{x} \cdot 5^{x} d x

Simplify

5^{x} \cdot 5^{x} : 2 5^{x}

= f \cdot \int_{3}^{4} 2 5^{x} d x

\int a^{x} d x = \frac{a ^{x}}{ln a}

= f [\frac{2 5 ^{x}}{ln ( 25 )}]_{3}^{4}

Simplify

ln (25) : 2 ln (5)

= f [\frac{2 5 ^{x}}{2 ln ( 5 )}]_{3}^{4}

Compute the boundaries:

\frac{187500}{ln ( 5 )}

= f \frac{187500}{ln ( 5 )}

Simplify

= \frac{187500 f}{ln ( 5 )}

What is the integral from 3 to 4 of f(5^x)5^x ?
The integral from 3 to 4 of f(5^x)5^x is (187500f)/(ln(5))