What is the integral of sqrt(x^2+a^2) ?

The integral of sqrt(x^2+a^2) is 1/2 xsqrt(x^2+a^2)+(a^2)/2 ln((|x+sqrt(a^2+x^2)|)/(|a|))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of sqrt(x^2+a^2)

integral

x^{2} + a^{2}

\frac{1}{2} x x^{2} + a^{2} + \frac{a ^{2}}{2} ln (\frac{x + a ^{2} + x ^{2}}{∣ a ∣}) + C

Solution steps

\int x^{2} + a^{2} d x

Apply Trigonometric Substitution

= \int a^{2} sec^{3} (u) d u

Take the constant out:

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

= a^{2} \cdot \int sec^{3} (u) d u

Apply Integral Reduction

= a^{2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

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

Substitute back

u = arctan (\frac{1}{a} x)

= a^{2} (\frac{sec ^{2} ( arctan ( \frac{1}{a} x ) ) sin ( arctan ( \frac{1}{a} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{a} x)) + sec (arctan (\frac{1}{a} x)))

Simplify

a^{2} (\frac{sec ^{2} ( arctan ( \frac{1}{a} x ) ) sin ( arctan ( \frac{1}{a} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{a} x)) + sec (arctan (\frac{1}{a} x))) : \frac{1}{2} x x^{2} + a^{2} + \frac{a ^{2}}{2} ln (\frac{x + a ^{2} + x ^{2}}{∣ a ∣})

= \frac{1}{2} x x^{2} + a^{2} + \frac{a ^{2}}{2} ln (\frac{x + a ^{2} + x ^{2}}{∣ a ∣})

Add a constant to the solution

= \frac{1}{2} x x^{2} + a^{2} + \frac{a ^{2}}{2} ln (\frac{x + a ^{2} + x ^{2}}{∣ a ∣}) + C

in t e g r a l \frac{y ^{2}}{2}

in t e g r a l \frac{x}{x ^{2} + a ^{2}}

in t e g r a l \frac{1}{( 2 - x )}

in t e g r a l \frac{( 1 - x ^{3} )}{( 1 - x )}

in t e g r a l \frac{1}{( 1 + lo g _{10} ( x ) )}

What is the integral of sqrt(x^2+a^2) ?
The integral of sqrt(x^2+a^2) is 1/2 xsqrt(x^2+a^2)+(a^2)/2 ln((|x+sqrt(a^2+x^2)|)/(|a|))+C