What is the integral of 1/(sqrt((x^2+a^2))) ?

The integral of 1/(sqrt((x^2+a^2))) is 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

Popular >

integral of 1/(sqrt((x^2+a^2)))

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

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

Solution steps

\int \frac{1}{x ^{2} + a ^{2}} d x

Apply Trigonometric Substitution

: \int sec (u) d u

= \int sec (u) d u

Use the common integral:

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

= ln ∣ tan (u) + sec (u) ∣

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{1}{x cos ( 1 - x )} d x

\int \frac{x ^{7}}{x ^{4} + 1} d x

\int x cos (2 x) sin (2 x) d x

\int \frac{5}{e ^{t}} d t

\int 0.05 e^{0.04 x} d x

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