What is the integral of 1/6 x5^{-1/3 x} ?

The integral of 1/6 x5^{-1/3 x} is -(5^{-x/3}x)/(2ln(5))-3/2*1/(ln^2(5))*5^{-x/3}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/6 x5^{-1/3 x}

\int \frac{1}{6} x 5^{- \frac{1}{3} x} d x

- \frac{5 ^{- \frac{x}{3}} x}{2 ln ( 5 )} - \frac{3}{2} \cdot \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- \frac{x}{3}} + C

Solution steps

\int \frac{1}{6} x \cdot 5^{- \frac{1}{3} x} d x

Take the constant out:

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

= \frac{1}{6} \cdot \int x \cdot 5^{- \frac{1}{3} x} d x

Apply u-substitution

: \int 9 \cdot 5^{u} u d u

= \frac{1}{6} \cdot \int 9 \cdot 5^{u} u d u

Take the constant out:

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

= \frac{1}{6} \cdot 9 \cdot \int 5^{u} u d u

Apply Integration By Parts

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

= \frac{1}{6} \cdot 9 (\frac{5 ^{u} u}{ln ( 5 )} - \int \frac{5 ^{u}}{ln ( 5 )} d u)

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

= \frac{1}{6} \cdot 9 (\frac{5 ^{u} u}{ln ( 5 )} - \frac{5 ^{u}}{ln ^{2} ( 5 )})

Substitute back

u = - \frac{1}{3} x

= \frac{1}{6} \cdot 9 (\frac{5 ^{- \frac{1}{3} x} ( - \frac{1}{3} x )}{ln ( 5 )} - \frac{5 ^{- \frac{1}{3} x}}{ln ^{2} ( 5 )})

Simplify

\frac{1}{6} \cdot 9 (\frac{5 ^{- \frac{1}{3} x} ( - \frac{1}{3} x )}{ln ( 5 )} - \frac{5 ^{- \frac{1}{3} x}}{ln ^{2} ( 5 )}) : - \frac{5 ^{- \frac{x}{3}} x}{2 ln ( 5 )} - \frac{3}{2} \cdot \frac{1}{ln ^{2} ( 5 )} \cdot 5^{- \frac{x}{3}}

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

Add a constant to the solution

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

\int \frac{cos ^{3} ( x )}{x} d x

\int 2 x y - tan (y) d x

\int \frac{4 x - arctan ^{3} ( x )}{1 + x ^{2}} d x

\int 4 x x + 2 d x

\int \frac{x ^{6}}{x ^{4} + 2 x ^{2} + 1} d x

What is the integral of 1/6 x5^{-1/3 x} ?
The integral of 1/6 x5^{-1/3 x} is -(5^{-x/3}x)/(2ln(5))-3/2*1/(ln^2(5))*5^{-x/3}+C