What is the integral of 1/2 x3^x ?

The integral of 1/2 x3^x is 1/2 ((3^xx)/(ln(3))-(3^x)/(ln^2(3)))+C

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integral of 1/2 x3^x

\int \frac{1}{2} x 3^{x} d x

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

Solution steps

\int \frac{1}{2} x \cdot 3^{x} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int x \cdot 3^{x} d x

Apply Integration By Parts

: \frac{3 ^{x} x}{ln ( 3 )} - \int \frac{3 ^{x}}{ln ( 3 )} d x

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

\int \frac{3 ^{x}}{ln ( 3 )} d x = \frac{3 ^{x}}{ln ^{2} ( 3 )}

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

Add a constant to the solution

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

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