What is the integral of 3/2 x2^x ?

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

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integral of 3/2 x2^x

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

: \frac{2 ^{x} x}{ln ( 2 )} - \int \frac{2 ^{x}}{ln ( 2 )} d x

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

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

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

Simplify

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

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

Add a constant to the solution

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

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