What is the integral of-x5^{-2x} ?

The integral of-x5^{-2x} is (25^{-x}x)/(2ln(5))+1/(4ln^2(5))*25^{-x}+C

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integral of-x5^{-2x}

\int - x 5^{- 2 x} d x

\frac{2 5 ^{- x} x}{2 ln ( 5 )} + \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x} + C

Solution steps

\int - x \cdot 5^{- 2 x} d x

Take the constant out:

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

= - \int x \cdot 5^{- 2 x} d x

Apply Integration By Parts

: - \frac{2 5 ^{- x} x}{2 ln ( 5 )} - \int - \frac{2 5 ^{- x}}{2 ln ( 5 )} d x

= - (- \frac{2 5 ^{- x} x}{2 ln ( 5 )} - \int - \frac{2 5 ^{- x}}{2 ln ( 5 )} d x)

\int - \frac{2 5 ^{- x}}{2 ln ( 5 )} d x = \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x}

= - (- \frac{2 5 ^{- x} x}{2 ln ( 5 )} - \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x})

Simplify

= \frac{2 5 ^{- x} x}{2 ln ( 5 )} + \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x}

Add a constant to the solution

= \frac{2 5 ^{- x} x}{2 ln ( 5 )} + \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x} + C

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What is the integral of-x5^{-2x} ?
The integral of-x5^{-2x} is (25^{-x}x)/(2ln(5))+1/(4ln^2(5))*25^{-x}+C