What is the integral of 1/(5^x) ?

The integral of 1/(5^x) is -1/(ln(5)*5^x)+C

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integral of 1/(5^x)

\int \frac{1}{5 ^{x}} d x

- \frac{1}{ln ( 5 ) \cdot 5 ^{x}} + C

Solution steps

\int \frac{1}{5 ^{x}} d x

Apply u-substitution

: \int \frac{1}{ln ( 5 ) u ^{2}} d u

= \int \frac{1}{ln ( 5 ) u ^{2}} d u

Take the constant out:

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

= \frac{1}{ln ( 5 )} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= \frac{1}{ln ( 5 )} (- \frac{1}{u})

Substitute back

u = 5^{x}

= \frac{1}{ln ( 5 )} (- \frac{1}{5 ^{x}})

Simplify

\frac{1}{ln ( 5 )} (- \frac{1}{5 ^{x}}) : - \frac{1}{ln ( 5 ) \cdot 5 ^{x}}

= - \frac{1}{ln ( 5 ) \cdot 5 ^{x}}

Add a constant to the solution

= - \frac{1}{ln ( 5 ) \cdot 5 ^{x}} + C

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