What is the integral of x2^{x+2} ?

The integral of x2^{x+2} is 4((2^xx)/(ln(2))-(2^x)/(ln^2(2)))+C

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integral of x2^{x+2}

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

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

Solution steps

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

x 2^{x + 2} = 2^{2} x 2^{x}

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

Simplify

= 4 \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

= 4 (\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 )}

= 4 (\frac{2 ^{x} x}{ln ( 2 )} - \frac{2 ^{x}}{ln ^{2} ( 2 )})

Add a constant to the solution

= 4 (\frac{2 ^{x} x}{ln ( 2 )} - \frac{2 ^{x}}{ln ^{2} ( 2 )}) + C

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What is the integral of x2^{x+2} ?
The integral of x2^{x+2} is 4((2^xx)/(ln(2))-(2^x)/(ln^2(2)))+C