Integration 0 to infinity (x^4 e^-2ax dx)

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You should evaluate the improper integral such that:

`int_0^(oo)x^4*e^(-2ax) dx = lim_(n->oo) int_0^n x^4*e^(-2ax) dx`

You need to use integration by parts such that:

`int udv = uv - int vdu`

`u = x^4 => du = 4x^3dx`

`dv = e^(-2ax) dx => v = (e^(-2ax))/(-2a)`

`int x^4*e^(-2ax) dx = (x^4*e^(-2ax))/(-2a) + int 4x^3(e^(-2ax))/(2a) dx`

`int x^4*e^(-2ax) dx = (x^4*e^(-2ax))/(-2a)+ (2/a)int x^3(e^(-2ax)) dx`

You need to use integration by parts to evaluate `int x^3(e^(-2ax)) dx `  such that:

`u =x^3 => du = 3x^2 dx`

`dv = e^(-2ax) dx => v = (e^(-2ax))/(-2a)`

`int x^3(e^(-2ax)) dx = (x^3*e^(-2ax))/(-2a) + int 3x^2(e^(-2ax))/(2a) dx`

`int x^3(e^(-2ax)) dx = (x^3*e^(-2ax))/(-2a) + (3/2a) int x^2(e^(-2ax)) dx`

You need to use integration by parts to evaluate `int x^2(e^(-2ax)) dx`  such that:

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

`dv = e^(-2ax) dx => v = (e^(-2ax))/(-2a)`

`int x^2(e^(-2ax)) dx = (x^2*e^(-2ax))/(-2a) + int 2x(e^(-2ax))/(2a) dx`

`int x^2(e^(-2ax)) dx = (x^2*e^(-2ax))/(-2a) + (1/a) int x(e^(-2ax)) dx`

You need to use integration by parts to evaluate `int x(e^(-2ax)) dx ` such that:

`u =x => du = dx`

`dv = e^(-2ax) dx => v = (e^(-2ax))/(-2a)`

`int x(e^(-2ax)) dx = (xe^(-2ax))/(-2a) + 1/(2a) int(e^(-2ax)) dx`

`int x(e^(-2ax)) dx = (xe^(-2ax))/(-2a)- 1/(2a)(e^(-2ax))/(2a)`

`int x^2(e^(-2ax)) dx = (x^2*e^(-2ax))/(-2a) + (1/a)((xe^(-2ax))/(-2a) - 1/(2a)(e^(-2ax))/(2a))`

`int x^3(e^(-2ax)) dx = (x^3*e^(-2ax))/(-2a) + (3/2a)((x^2*e^(-2ax))/(-2a) + (1/a)((xe^(-2ax))/(-2a) - 1/(2a)(e^(-2ax))/(2a)))`

`int x^4*e^(-2ax) dx = (x^4*e^(-2ax))/(-2a) + (2/a)((x^3*e^(-2ax))/(-2a) + (3/2a)((x^2*e^(-2ax))/(-2a) + (1/a)((xe^(-2ax))/(-2a) - 1/(2a)(e^(-2ax))/(2a))))`

Hence, evaluating the definite integral using the fundamental theorem of calculus yields:

`int_0^n x^4*e^(-2ax) dx = ((x^4*e^(-2ax))/(-2a) + (2/a)((x^3*e^(-2ax))/(-2a) + (3/2a)((x^2*e^(-2ax))/(-2a) + (1/a)((xe^(-2ax))/(-2a) - 1/(2a)(e^(-2ax))/(2a)))))|_0^n`

`int_0^n x^4*e^(-2ax) dx = ((n^4*e^(-2an))/(-2a) + (2/a)((n^3*e^(-2an))/(-2a) + (3/2a)((n^2*e^(-2an))/(-2a) + (1/a)((ne^(-2an))/(-2a) - 1/(2a)(e^(-2an))/(2a))))+ 6/(8a^5))`

Hence, evaluating the improper integral yields:

`lim_(n->oo) int_0^n x^4*e^(-2ax) dx = lim_(n->oo) ((n^4*e^(-2an))/(-2a) + (2/a)((n^3*e^(-2an))/(-2a) + (3/2a)((n^2*e^(-2an))/(-2a) + (1/a)((ne^(-2an))/(-2a) - 1/(2a)(e^(-2an))/(2a)))) - 6/(8a^5)) = 0+6/(8a^5)`

`lim_(n->oo) int_0^n x^4*e^(-2ax) dx = 3/(4a^5)`

Hence, evaluating the given improper integral yields `lim_(n->oo) int_0^n x^4*e^(-2ax) dx = 3/(4a^5).`

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