What is the integral of-x6^{-2/3 x} ?

The integral of-x6^{-2/3 x} is -1/(ln^2(6))*3^{(-2x+3)/3}*2^{(-2x-6)/3}(-2ln(6)x-3)+C

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integral of-x6^{-2/3 x}

\int - x 6^{- \frac{2}{3} x} d x

- \frac{1}{ln ^{2} ( 6 )} \cdot 3^{\frac{- 2 x + 3}{3}} \cdot 2^{\frac{- 2 x - 6}{3}} (- 2 ln (6) x - 3) + C

Solution steps

\int - x \cdot 6^{- \frac{2}{3} x} d x

Take the constant out:

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

= - \int x \cdot 6^{- \frac{2}{3} x} d x

Apply u-substitution

: \int \frac{9 ln ( u )}{4 ln ^{2} ( 6 )} d u

= - \int \frac{9 ln ( u )}{4 ln ^{2} ( 6 )} d u

Take the constant out:

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

= - \frac{9}{4 ln ^{2} ( 6 )} \cdot \int ln (u) d u

Apply Integration By Parts

: u ln (u) - \int 1 d u

= - \frac{9}{4 ln ^{2} ( 6 )} (u ln (u) - \int 1 d u)

\int 1 d u = u

= - \frac{9}{4 ln ^{2} ( 6 )} (u ln (u) - u)

Substitute back

u = 6^{- \frac{2}{3} x}

= - \frac{9}{4 ln ^{2} ( 6 )} (6^{- \frac{2}{3} x} ln (6^{- \frac{2}{3} x}) - 6^{- \frac{2}{3} x})

Simplify

- \frac{9}{4 ln ^{2} ( 6 )} (6^{- \frac{2}{3} x} ln (6^{- \frac{2}{3} x}) - 6^{- \frac{2}{3} x}) : - \frac{1}{ln ^{2} ( 6 )} \cdot 3^{\frac{- 2 x + 3}{3}} \cdot 2^{\frac{- 2 x - 6}{3}} (- 2 ln (6) x - 3)

= - \frac{1}{ln ^{2} ( 6 )} \cdot 3^{\frac{- 2 x + 3}{3}} \cdot 2^{\frac{- 2 x - 6}{3}} (- 2 ln (6) x - 3)

Add a constant to the solution

= - \frac{1}{ln ^{2} ( 6 )} \cdot 3^{\frac{- 2 x + 3}{3}} \cdot 2^{\frac{- 2 x - 6}{3}} (- 2 ln (6) x - 3) + C

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What is the integral of-x6^{-2/3 x} ?
The integral of-x6^{-2/3 x} is -1/(ln^2(6))*3^{(-2x+3)/3}*2^{(-2x-6)/3}(-2ln(6)x-3)+C