IntegralHow to determine integral of a rational function if the numerator is ln of variable and denominator is square root of the same variable?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Considering the imposed constraints, we'll take the following rational function:

f(x) = ln x/sqrt x

We'll integrate using parts.

Int udv = uv - Int vdu

We'll put u = ln x.

We'll differentiate both sides:

du = dx/x

We'll put dv = 1/sqrtx

We'll integrate both sides;

Int dv = Int dx/sqrtx

v = 2sqrt x

We'll substitute u,v,du and dv in the formula above:

Int (ln x/sqrt x)dx = 2*(ln x)*(sqrt x) - 2 Int (sqrtx/x) dx

But sqrtx/x = sqrtx/(sqrtx)*(sqrt x)

sqrtx/x = 1/sqrt x

We know that Int dx/sqrtx = 2sqrt x

Int (ln x/sqrt x)dx = 2*(ln x)*(sqrt x) - 2*2sqrt x + C

Int (ln x/sqrt x)dx = 2*(ln x)*(sqrt x) - 4*sqrt x + C

Int (ln x/sqrt x)dx = 2(sqrt x)( ln x - 2) + C

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