step by step integral x^2*2^x*dx

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should perform integration by parts using the following formula such that:

`int udv = uv - int vdu`

`u = x^2 => du = 2xdx`

`dv = 2^x dx => v = 2^x/ln 2`

Reasoning by analogy yields:

`int x^2*2^x dx = x^2*2^x/ln 2 - int 2^x/ln 2*2xdx`

`int x^2*2^x dx = x^2*2^x/ln 2 - (2/ln 2)int x*2^x ` dx

Performing integration by parts for int `x*2^x dx`  yields:

`u = x => du = dx`

`dv = 2^xdx => v = 2^x/ln 2`

`int x*2^x dx = x*2^x/ln 2 - int2^x/ln 2 dx`

`int x*2^x dx = x*2^x/ln 2 - (1/ln 2) int 2^x dx`

`int x*2^x dx = x*2^x/ln 2 - 2^x/(ln^2 2) + c`

Substituting `x*2^x/ln 2 - 2^x/(ln^2 2)`  for `int x*2^x dx`  yields:

`int x^2*2^x dx = x^2*2^x/ln 2 - (2/ln 2)(x*2^x/ln 2 - 2^x/(ln^2 2))+c`

`int x^2*2^x dx = x^2*2^x/ln 2 - (x*2^(x+1))/(ln^2 2) + (2^(x+1))/(ln^3 2) + c`

Hence, evaluating the integral using parts yields `int x^2*2^x dx = x^2*2^x/ln 2 - (x*2^(x+1))/(ln^2 2) + (2^(x+1))/(ln^3 2) + c.`

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