What is the integral of (x*9^{2x^2-8}) ?

The integral of (x*9^{2x^2-8}) is (3^{4x^2-16})/(8ln(3))+C

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integral of (x*9^{2x^2-8})

\int (x \cdot 9^{2 x^{2} - 8}) d x

\frac{3 ^{4 x^{2} - 16}}{8 ln ( 3 )} + C

Solution steps

\int x \cdot 9^{2 x^{2} - 8} d x

x 9^{2 x^{2} - 8} = 9^{- 8} x 9^{2 x^{2}}

= \int 9^{- 8} x \cdot 9^{2 x^{2}} d x

Take the constant out:

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

= 9^{- 8} \cdot \int x \cdot 9^{2 x^{2}} d x

9^{- 8} = \frac{1}{43046721}

= \frac{1}{43046721} \cdot \int x \cdot 9^{2 x^{2}} d x

Apply u-substitution

: \int \frac{8 1 ^{u}}{2} d u

= \frac{1}{43046721} \cdot \int \frac{8 1 ^{u}}{2} d u

Take the constant out:

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

= \frac{1}{43046721} \cdot \frac{1}{2} \cdot \int 8 1^{u} d u

\int a^{x} d x = \frac{a ^{x}}{ln a}

= \frac{1}{43046721} \cdot \frac{1}{2} \cdot \frac{8 1 ^{u}}{ln ( 81 )}

Substitute back

u = x^{2}

= \frac{1}{43046721} \cdot \frac{1}{2} \cdot \frac{8 1 ^{x^{2}}}{ln ( 81 )}

Simplify

\frac{1}{43046721} \cdot \frac{1}{2} \cdot \frac{8 1 ^{x^{2}}}{ln ( 81 )} : \frac{3 ^{4 x^{2} - 16}}{8 ln ( 3 )}

= \frac{3 ^{4 x^{2} - 16}}{8 ln ( 3 )}

Add a constant to the solution

= \frac{3 ^{4 x^{2} - 16}}{8 ln ( 3 )} + C

\frac{\partial}{\partial x} (1 + x ln (x y - 9))

\frac{2 s + 2}{s ^{2} + 2 s + 5}

\frac{\partial}{\partial y} (\frac{x}{1 + y ^{2}})

16 + t

\frac{d}{d x} ((5 x + 2)^{4} (3 x - 7))

What is the integral of (x*9^{2x^2-8}) ?
The integral of (x*9^{2x^2-8}) is (3^{4x^2-16})/(8ln(3))+C