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Popular >

integral of x/y+y/x

Solution

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

Solution

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

Popular Examples

integral from 0 to 2 of x^3sqrt(1+9x^4)

\int_{0}^{2} x^{3} 1 + 9 x^{4} d x

derivative of 8arctan(x)

\frac{d}{d x} (8 arctan (x))

integral from 1 to e of 1/x

\int_{1}^{e} \frac{1}{x} d x

derivative of y=ln(1/(xsqrt(x+1)))derivative of

y = ln (\frac{1}{x x + 1})

x y^{^{'}} - 2 y - 16 = 0

Frequently Asked Questions (FAQ)

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