What is the integral from x to 2x of y/(x^2+y^2) ?

The integral from x to 2x of y/(x^2+y^2) is 1/2 (ln(5)-ln(2))

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integral from x to 2x of y/(x^2+y^2)

\int_{x}^{2 x} \frac{y}{x ^{2} + y ^{2}} d y

\frac{1}{2} (ln (5) - ln (2))

Decimal

0.45814 \dots

Solution steps

\int_{x}^{2 x} \frac{y}{x ^{2} + y ^{2}} d y

Apply u-substitution

= \int_{2 x^{2}}^{5 x^{2}} \frac{1}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{2 x^{2}}^{5 x^{2}} \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} [ln ∣ u ∣]_{2 x^{2}}^{5 x^{2}}

Compute the boundaries:

ln (5) - ln (2)

= \frac{1}{2} (ln (5) - ln (2))

What is the integral from x to 2x of y/(x^2+y^2) ?
The integral from x to 2x of y/(x^2+y^2) is 1/2 (ln(5)-ln(2))