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

The integral of x^2*2^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^2*2^x

\int x^{2} \cdot 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

f (x) = x + x^{\frac{2}{3}} - 3 x

x \to 0 lim (2 sin (x) ln (x))

2 e^{x} - e^{x + 1} + e^{\frac{2}{3}} - 5 π^{2}

x \to 0 + lim (sin^{2} (x))

n = 0 \sum \infty (\frac{3}{2})^{n}

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