What is the integral of x^22^x ?

The integral of x^22^x is (2^xx^2)/(ln(2))-1/(ln(2))((2^{x+1}x)/(ln(2))-1/(ln^2(2))*2^{1+x})+C

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integral of x^22^x

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

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

Solution steps

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

Apply Integration By Parts

: \frac{2 ^{x} x ^{2}}{ln ( 2 )} - \int \frac{2 ^{x + 1} x}{ln ( 2 )} d x

= \frac{2 ^{x} x ^{2}}{ln ( 2 )} - \int \frac{2 ^{x + 1} x}{ln ( 2 )} d x

\int \frac{2 ^{x + 1} x}{ln ( 2 )} d x = \frac{1}{ln ( 2 )} (\frac{2 ^{x + 1} x}{ln ( 2 )} - \frac{1}{ln ^{2} ( 2 )} \cdot 2^{1 + x})

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

Add a constant to the solution

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

\int_{1}^{7} - \frac{1}{3} y^{2} + 3 y d y

x \frac{d y}{d x} + x y = 1 - y

n = 0 \sum \infty \frac{x ^{n}}{3 ^{n}}

x^{2} y y^{^{'}} + 3 x y^{2} = 0

y = 5 x - 4 x^{2}, (- 2, - 26)

What is the integral of x^22^x ?
The integral of x^22^x is (2^xx^2)/(ln(2))-1/(ln(2))((2^{x+1}x)/(ln(2))-1/(ln^2(2))*2^{1+x})+C