What is the integral of (x*2^x) ?

The integral of (x*2^x) is (2^xx)/(ln(2))-(2^x)/(ln^2(2))+C

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integral of (x*2^x)

\int (x \cdot 2^{x}) d x

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

Solution steps

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

Add a constant to the solution

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

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