What is the integral from x to 2x of x/y ?

The integral from x to 2x of x/y is ln(2)x

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral from x to 2x of x/y

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

ln (2) x

Solution steps

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{y} d y = ln (∣ y ∣)

= x [ln ∣ y ∣]_{x}^{2 x}

Compute the boundaries:

ln (2 ∣ x ∣) - ln ∣ x ∣

= x (ln (2 ∣ x ∣) - ln ∣ x ∣)

Simplify

= ln (2) x

\int_{0.4}^{0.6} \frac{5}{3} d x

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

\int_{0}^{1} ln (1 + x^{2}) x d x

\int_{0}^{1} 1 - t d t

\int_{0}^{3} (9 - x^{2}) d x

What is the integral from x to 2x of x/y ?
The integral from x to 2x of x/y is ln(2)x