What is the integral of 5/2 x5^{-x+6} ?

The integral of 5/2 x5^{-x+6} is 78125/2 (-(5^{-x}x)/(ln(5))-1/(ln^2(5))*5^{-x})+C

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integral of 5/2 x5^{-x+6}

\int \frac{5}{2} x 5^{- x + 6} d x

\frac{78125}{2} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x}) + C

Solution steps

\int \frac{5}{2} x \cdot 5^{- x + 6} d x

Take the constant out:

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

= \frac{5}{2} \cdot \int x \cdot 5^{- x + 6} d x

x 5^{- x + 6} = 5^{6} x 5^{- x}

= \frac{5}{2} \cdot \int 5^{6} x \cdot 5^{- x} d x

Take the constant out:

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

= \frac{5}{2} \cdot 5^{6} \cdot \int x \cdot 5^{- x} d x

Simplify

= \frac{5}{2} \cdot 15625 \cdot \int x \cdot 5^{- x} d x

Apply Integration By Parts

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

= \frac{5}{2} \cdot 15625 (- \frac{5 ^{- x} x}{ln ( 5 )} - \int - \frac{5 ^{- x}}{ln ( 5 )} d x)

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

= \frac{5}{2} \cdot 15625 (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x})

Simplify

\frac{5}{2} \cdot 15625 (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x}) : \frac{78125}{2} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x})

= \frac{78125}{2} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x})

Add a constant to the solution

= \frac{78125}{2} (- \frac{5 ^{- x} x}{ln ( 5 )} - \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- x}) + C

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What is the integral of 5/2 x5^{-x+6} ?
The integral of 5/2 x5^{-x+6} is 78125/2 (-(5^{-x}x)/(ln(5))-1/(ln^2(5))*5^{-x})+C