What is the integral of-3/2 x5^{-2x+1} ?

The integral of-3/2 x5^{-2x+1} is -15/2 (-(25^{-x}x)/(2ln(5))-1/(4ln^2(5))*25^{-x})+C

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integral of-3/2 x5^{-2x+1}

\int - \frac{3}{2} x 5^{- 2 x + 1} d x

- \frac{15}{2} (- \frac{2 5 ^{- x} x}{2 ln ( 5 )} - \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x}) + C

Solution steps

\int - \frac{3}{2} x \cdot 5^{- 2 x + 1} d x

Take the constant out:

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

= - \frac{3}{2} \cdot \int x \cdot 5^{- 2 x + 1} d x

x 5^{- 2 x + 1} = 5^{1} x 5^{- 2 x}

= - \frac{3}{2} \cdot \int 5^{1} 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

= - \frac{3}{2} \cdot 5^{1} \cdot \int x \cdot 5^{- 2 x} d x

Simplify

= - \frac{3}{2} \cdot 5 \cdot \int x \cdot 5^{- 2 x} d x

Apply Integration By Parts

= - \frac{3}{2} \cdot 5 (- \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{3}{2} \cdot 5 (- \frac{2 5 ^{- x} x}{2 ln ( 5 )} - \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x})

Simplify

- \frac{3}{2} \cdot 5 (- \frac{2 5 ^{- x} x}{2 ln ( 5 )} - \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x}) : - \frac{15}{2} (- \frac{2 5 ^{- x} x}{2 ln ( 5 )} - \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x})

= - \frac{15}{2} (- \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{15}{2} (- \frac{2 5 ^{- x} x}{2 ln ( 5 )} - \frac{1}{4 ln ^{2} ( 5 )} \cdot 2 5^{- x}) + C

What is the integral of-3/2 x5^{-2x+1} ?
The integral of-3/2 x5^{-2x+1} is -15/2 (-(25^{-x}x)/(2ln(5))-1/(4ln^2(5))*25^{-x})+C