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

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

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integral of (x^2+a^2)^{1/2}

integral

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

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

Solution steps

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

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

Add a constant to the solution

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

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