What is the integral of (sqrt(x^2-25))/(2x^2) ?

The integral of (sqrt(x^2-25))/(2x^2) is 1/2 (ln(1/5 |sqrt(x^2-25)+x|)-(sqrt(x^2-25))/x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sqrt(x^2-25))/(2x^2)

\int \frac{x ^{2} - 25}{2 x ^{2}} d x

\frac{1}{2} (ln (\frac{1}{5} x^{2} - 25 + x) - \frac{x ^{2} - 25}{x}) + C

Solution steps

\int \frac{x ^{2} - 25}{2 x ^{2}} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{x ^{2} - 25}{x ^{2}} d x

Apply Trigonometric Substitution

: \int tan^{2} (u) cos (u) d u

= \frac{1}{2} \cdot \int tan^{2} (u) cos (u) d u

Simplify

tan^{2} (u) cos (u) : sin (u) tan (u)

= \frac{1}{2} \cdot \int sin (u) tan (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int \frac{1 - cos ^{2} ( u )}{cos ( u )} d u

Expand

\frac{1 - cos ^{2} ( u )}{cos ( u )} : \frac{1}{cos ( u )} - cos (u)

= \frac{1}{2} \cdot \int \frac{1}{cos ( u )} - cos (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int \frac{1}{cos ( u )} d u - \int cos (u) d u)

\int \frac{1}{cos ( u )} d u = ln ∣ tan (u) + sec (u) ∣

\int cos (u) d u = sin (u)

= \frac{1}{2} (ln ∣ tan (u) + sec (u) ∣ - sin (u))

Substitute back

u = arcsec (\frac{1}{5} x)

= \frac{1}{2} (ln tan (arcsec (\frac{1}{5} x)) + sec (arcsec (\frac{1}{5} x)) - sin (arcsec (\frac{1}{5} x)))

Simplify

\frac{1}{2} (ln tan (arcsec (\frac{1}{5} x)) + sec (arcsec (\frac{1}{5} x)) - sin (arcsec (\frac{1}{5} x))) : \frac{1}{2} (ln (\frac{1}{5} x^{2} - 25 + x) - \frac{x ^{2} - 25}{x})

= \frac{1}{2} (ln (\frac{1}{5} x^{2} - 25 + x) - \frac{x ^{2} - 25}{x})

Add a constant to the solution

= \frac{1}{2} (ln (\frac{1}{5} x^{2} - 25 + x) - \frac{x ^{2} - 25}{x}) + C

\int x^{2} \cdot x^{3} + 3 d x

\frac{d y}{d x} = \frac{( 9 x ^{11} x ^{23} )}{y}

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

\int_{- 2}^{3} \frac{8}{x ^{4}} d x

\int \frac{400}{x ^{3} - 20 x ^{2}} d x

What is the integral of (sqrt(x^2-25))/(2x^2) ?
The integral of (sqrt(x^2-25))/(2x^2) is 1/2 (ln(1/5 |sqrt(x^2-25)+x|)-(sqrt(x^2-25))/x)+C