What is the integral of x/(10^x) ?

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

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integral of x/(10^x)

\int \frac{x}{1 0 ^{x}} d x

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

Solution steps

\int \frac{x}{1 0 ^{x}} d x

Apply exponent rule:

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

\frac{x}{1 0 ^{x}} = (x) 1 0^{- x}

= \int x \cdot 1 0^{- x} d x

Apply Integration By Parts

: - \frac{1 0 ^{- x} x}{ln ( 10 )} - \int - \frac{1 0 ^{- x}}{ln ( 10 )} d x

= - \frac{1 0 ^{- x} x}{ln ( 10 )} - \int - \frac{1 0 ^{- x}}{ln ( 10 )} d x

\int - \frac{1 0 ^{- x}}{ln ( 10 )} d x = \frac{1}{ln ^{2} ( 10 )} \cdot 1 0^{- x}

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

Add a constant to the solution

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

10 - 7 x, x^{2} - 20, 0

\frac{d}{d x} (x^{6} f (x))

f (u) = ln (u)

\int_{0}^{1} (- 12 e^{x}) d x

x \to \infty lim (\frac{- 1}{x})

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