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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from x to x^2 of y/x

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

\frac{x ( x ^{2} - 1 )}{2}

Solution steps

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

Take the constant out:

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

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

Apply the Power Rule

= \frac{1}{x} [\frac{y ^{2}}{2}]_{x}^{x^{2}}

Simplify

\frac{1}{x} [\frac{y ^{2}}{2}]_{x}^{x^{2}} : \frac{[ \frac{y ^{2}}{2} ] _{x}^{x^{2}}}{x}

= \frac{[ \frac{y ^{2}}{2} ] _{x}^{x^{2}}}{x}

Compute the boundaries:

\frac{x ^{4}}{2} - \frac{x ^{2}}{2}

= \frac{\frac{x ^{4}}{2} - \frac{x ^{2}}{2}}{x}

Simplify

= \frac{x ( x ^{2} - 1 )}{2}

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