# evaluate the following definite/indefinite integral: ||   x^3e^x+4+(ln(x^2+3)/2x) dx

Since the problem does not provide the limits of integration, hence you need to evaluate the indefinite integral such that:

int (x^3e^x+4+ ln((x^2+3)/(2x))) dx

You need to split the integral into three simpler integrals to make the process of evaluation easier such that:

int (x^3e^x+4+ ln((x^2+3)/(2x))) dx =int (x^3e^x)dx + int 4dx + int ln((x^2+3)/(2x)) dx

You need to evaluate the first integral using the formula of integration by parts such that:

int udv = uv - int vdu

You should come up with the substitution such that:

u = x^3 =gt du = 3x^2 dx

dv = e^x dx =gt v = e^x

int (x^3e^x)dx = (x^3e^x) - int (3x^2e^x)dx

u = x^2 =gt du = 2xdx

v = e^x

int (3x^2e^x)dx = 3e^x*x^2 - 6int xe^x dx

u = x =gt du = dx

v = e^x

6int xe^x dx = 6xe^x - 6int e^x

6int xe^x dx = 6xe^x - 6 e^x

int (3x^2e^x)dx = 3e^x*x^2 - 6xe^x+ 6 e^x

int (x^3e^x)dx = x^3*e^x - 3e^x*x^2+ 6xe^x- 6 e^x + c

You need to evaluate the second integral such that:

int 4dx = 4x + c

You need to evaluate the third integral using integration by parts method such that:

int ln((x^2+3)/(2x)) dx

u = ln ((x^2+3)/(2x)) =gt du = (2x)/(x^2+3)*(4x^2 - 2x^2- 6)/(4x^2) dx

du = (2(x^2-3))/(x(x^2+3))

dv = dx =gt v = x

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

You need to evaluate int (2(x^2-3))/(x^2+3)  such that:

2int (x^2-3)/(x^2+3)dx = 2int (x^2 + 3 - 6)/(x^2+3)dx

2int (x^2 + 3 - 6)/(x^2+3)dx = 2int (x^2 + 3)/(x^2+3)dx - 12int (dx)/(x^2+3)

2int (x^2 + 3 - 6)/(x^2+3)dx = 2x - 12/sqrt3*arctan (x/sqrt3) + c

Hence, evaluating the given integral yields int (x^3e^x+4+ ln((x^2+3)/(2x)))dx = x^3*e^x - 3e^x*x^2 + 6xe^x - 6 e^x + 4x + 2x - 12/sqrt3*arctan (x/sqrt3) + c .

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