What is the integral of (x+1)2^x ?

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

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integral of (x+1)2^x

\int (x + 1) 2^{x} d x

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

Solution steps

\int (x + 1) \cdot 2^{x} d x

Expand

(x + 1) \cdot 2^{x} : 2^{x} x + 2^{x}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Add a constant to the solution

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

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