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

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

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

\int (2 - x) 5^{(2 - x)^{2}} d x

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

Solution steps

\int (2 - x) \cdot 5^{(2 - x)^{2}} d x

Apply u-substitution

: \int - 5^{u^{2}} u d u

= \int - 5^{u^{2}} u d u

Take the constant out:

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

= - \int 5^{u^{2}} u d u

Apply u-substitution

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

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

Integral of a constant:

\int a d x = a x

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

Substitute back

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

Add a constant to the solution

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

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