What is the integral of 1/(10^zln(10)) ?

The integral of 1/(10^zln(10)) is -1/(ln^2(10)*10^z)+C

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integral of 1/(10^zln(10))

\int \frac{1}{1 0 ^{z} ln ( 10 )}

- \frac{1}{ln ^{2} ( 10 ) \cdot 1 0 ^{z}} + C

Solution steps

\int \frac{1}{1 0 ^{z} ln ( 10 )} d z

Take the constant out:

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

= \frac{1}{ln ( 10 )} \cdot \int \frac{1}{1 0 ^{z}} d z

Apply u-substitution

: \int \frac{1}{ln ( 10 ) u ^{2}} d u

= \frac{1}{ln ( 10 )} \cdot \int \frac{1}{ln ( 10 ) 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 ( 10 )} \cdot \frac{1}{ln ( 10 )} \cdot \int \frac{1}{u ^{2}} d u

Apply the Power Rule

: - \frac{1}{u}

= \frac{1}{ln ( 10 )} \cdot \frac{1}{ln ( 10 )} (- \frac{1}{u})

Substitute back

u = 1 0^{z}

= \frac{1}{ln ( 10 )} \cdot \frac{1}{ln ( 10 )} (- \frac{1}{1 0 ^{z}})

Simplify

\frac{1}{ln ( 10 )} \cdot \frac{1}{ln ( 10 )} (- \frac{1}{1 0 ^{z}}) : - \frac{1}{ln ^{2} ( 10 ) \cdot 1 0 ^{z}}

= - \frac{1}{ln ^{2} ( 10 ) \cdot 1 0 ^{z}}

Add a constant to the solution

= - \frac{1}{ln ^{2} ( 10 ) \cdot 1 0 ^{z}} + C

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What is the integral of 1/(10^zln(10)) ?
The integral of 1/(10^zln(10)) is -1/(ln^2(10)*10^z)+C