What is the integral of (x/y+y/x) ?

The integral of (x/y+y/x) is xln|y|+(y^2)/(2x)+C

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integral of (x/y+y/x)

\int (\frac{x}{y} + \frac{y}{x}) d y

x ln ∣ y ∣ + \frac{y ^{2}}{2 x} + C

Solution steps

\int \frac{x}{y} + \frac{y}{x} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{y} d y + \int \frac{y}{x} d y

\int \frac{x}{y} d y = x ln ∣ y ∣

\int \frac{y}{x} d y = \frac{y ^{2}}{2 x}

= x ln ∣ y ∣ + \frac{y ^{2}}{2 x}

Add a constant to the solution

= x ln ∣ y ∣ + \frac{y ^{2}}{2 x} + C

y = \frac{2 x - 3}{3 x + 1}

1 + 4 x

\frac{d}{d x} ((4^{\frac{2}{3}} - x^{\frac{2}{3}})^{\frac{3}{2}})

\int_{2}^{4} 5 t^{2} d t

\frac{d}{d x} ((2 x - 3)^{2} (5 x^{2} + 2)^{3})

What is the integral of (x/y+y/x) ?
The integral of (x/y+y/x) is xln|y|+(y^2)/(2x)+C