What is the integral of x/(5^{x^2)} ?

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

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integral of x/(5^{x^2)}

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

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

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

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

Apply u-substitution

: \int - \frac{1}{2 ln ( 5 )} d u

= \int - \frac{1}{2 ln ( 5 )} d u

Integral of a constant:

\int a d x = a x

= (- \frac{1}{2 ln ( 5 )}) u

Substitute back

u = 5^{- x^{2}}

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

Simplify

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

Add a constant to the solution

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

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