What is the integral of (10)/(5^{3x)} ?

The integral of (10)/(5^{3x)} is -(2*5^{-3x+1})/(3ln(5))+C

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integral of (10)/(5^{3x)}

\int \frac{10}{5 ^{3 x}} d x

- \frac{2 \cdot 5 ^{- 3 x + 1}}{3 ln ( 5 )} + C

Solution steps

\int \frac{10}{5 ^{3 x}} d x

Take the constant out:

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

= 10 \cdot \int \frac{1}{5 ^{3 x}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{5 ^{3 x}} = 5^{- 3 x}

= 10 \cdot \int 5^{- 3 x} d x

Simplify

5^{- 3 x} : 12 5^{- x}

= 10 \cdot \int 12 5^{- x} d x

Apply u-substitution

= 10 \cdot \int - 12 5^{u} d u

Take the constant out:

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

= 10 (- \int 12 5^{u} d u)

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

= 10 (- \frac{12 5 ^{u}}{ln ( 125 )})

Substitute back

u = - x

= 10 (- \frac{12 5 ^{- x}}{ln ( 125 )})

Simplify

10 (- \frac{12 5 ^{- x}}{ln ( 125 )}) : - \frac{2 \cdot 5 ^{- 3 x + 1}}{3 ln ( 5 )}

= - \frac{2 \cdot 5 ^{- 3 x + 1}}{3 ln ( 5 )}

Add a constant to the solution

= - \frac{2 \cdot 5 ^{- 3 x + 1}}{3 ln ( 5 )} + C

What is the integral of (10)/(5^{3x)} ?
The integral of (10)/(5^{3x)} is -(2*5^{-3x+1})/(3ln(5))+C