What is the integral of 5^{3-2x^2}x ?

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

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integral of 5^{3-2x^2}x

\int 5^{3 - 2 x^{2}} x d x

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

Solution steps

\int 5^{3 - 2 x^{2}} x d x

5^{3 - 2 x^{2}} x = 5^{3} \cdot 5^{- 2 x^{2}} x

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

Take the constant out:

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

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

Simplify

= 125 \cdot \int 5^{- 2 x^{2}} x d x

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

= 125 \cdot \int 2 5^{- x^{2}} x d x

Apply u-substitution

= 125 \cdot \int - \frac{1}{4 ln ( 5 )} d u

Integral of a constant:

\int a d x = a x

= 125 (- \frac{1}{4 ln ( 5 )}) u

Substitute back

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

= 125 (- \frac{1}{4 ln ( 5 )}) \cdot 2 5^{- x^{2}}

Simplify

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

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

Add a constant to the solution

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

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