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

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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