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

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

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integral of 2x*2^x

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

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

Solution steps

\int 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

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

Apply Integration By Parts

= 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 )}

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

Add a constant to the solution

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

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