What is the integral of x3^x2^{2x} ?

The integral of x3^x2^{2x} is 1/(ln^2(12))*12^x+(12^xx)/(ln(12))-1/(ln^2(12))*3^x*2^{1+2x}+C

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integral of x3^x2^{2x}

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

\frac{1}{ln ^{2} ( 12 )} \cdot 1 2^{x} + \frac{1 2 ^{x} x}{ln ( 12 )} - \frac{1}{ln ^{2} ( 12 )} \cdot 3^{x} \cdot 2^{1 + 2 x} + C

Solution steps

\int x \cdot 3^{x} \cdot 2^{2 x} d x

Apply Integration By Parts

: 1 2^{x} x^{2} - \int x (ln (12) \cdot 1 2^{x} x + 1 2^{x}) d x

= 1 2^{x} x^{2} - \int x (ln (12) \cdot 1 2^{x} x + 1 2^{x}) d x

\int x (ln (12) \cdot 1 2^{x} x + 1 2^{x}) d x = 1 2^{x} x^{2} - \frac{1}{ln ^{2} ( 12 )} \cdot 1 2^{x} - \frac{1 2 ^{x} x}{ln ( 12 )} + \frac{1}{ln ^{2} ( 12 )} \cdot 3^{x} \cdot 2^{1 + 2 x}

= 1 2^{x} x^{2} - (1 2^{x} x^{2} - \frac{1}{ln ^{2} ( 12 )} \cdot 1 2^{x} - \frac{1 2 ^{x} x}{ln ( 12 )} + \frac{1}{ln ^{2} ( 12 )} \cdot 3^{x} \cdot 2^{1 + 2 x})

Simplify

= \frac{1}{ln ^{2} ( 12 )} \cdot 1 2^{x} + \frac{1 2 ^{x} x}{ln ( 12 )} - \frac{1}{ln ^{2} ( 12 )} \cdot 3^{x} \cdot 2^{1 + 2 x}

Add a constant to the solution

= \frac{1}{ln ^{2} ( 12 )} \cdot 1 2^{x} + \frac{1 2 ^{x} x}{ln ( 12 )} - \frac{1}{ln ^{2} ( 12 )} \cdot 3^{x} \cdot 2^{1 + 2 x} + C

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What is the integral of x3^x2^{2x} ?
The integral of x3^x2^{2x} is 1/(ln^2(12))*12^x+(12^xx)/(ln(12))-1/(ln^2(12))*3^x*2^{1+2x}+C