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))

Popular >

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

= \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))

\int_{- \infty}^{0} x e^{5 x + 2} d x

\int_{0}^{3} \frac{x}{16 + x ^{2}} d x

\int_{1}^{3} x^{2} - 1 d x

\int_{0}^{5} 4 x d x

\int_{1}^{3} \frac{x ^{2} + 2}{x} d x

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))