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

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

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integral of sqrt(a+x^2)

\int a + x^{2} d x

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

Solution steps

\int a + x^{2} d x

Apply Trigonometric Substitution

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

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

Take the constant out:

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

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

Apply Integral Reduction

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

Substitute back

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

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

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

Add a constant to the solution

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

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