What is the integral of sqrt(81+x^2) ?

The integral of sqrt(81+x^2) is 81(1/162 xsqrt(81+x^2)+1/2 ln((|x+sqrt(81+x^2)|)/9))+C

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integral of sqrt(81+x^2)

\int 81 + x^{2} d x

81 (\frac{1}{162} x 81 + x^{2} + \frac{1}{2} ln (\frac{x + 81 + x ^{2}}{9})) + C

Solution steps

\int 81 + x^{2} d x

Apply Trigonometric Substitution

: \int 81 sec^{3} (u) d u

= \int 81 sec^{3} (u) d u

Take the constant out:

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

= 81 \cdot \int sec^{3} (u) d u

Apply Integral Reduction

= 81 (\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) ∣

= 81 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arctan (\frac{1}{9} x)

= 81 (\frac{sec ^{2} ( arctan ( \frac{1}{9} x ) ) sin ( arctan ( \frac{1}{9} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{9} x)) + sec (arctan (\frac{1}{9} x)))

Simplify

81 (\frac{sec ^{2} ( arctan ( \frac{1}{9} x ) ) sin ( arctan ( \frac{1}{9} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{9} x)) + sec (arctan (\frac{1}{9} x))) : 81 (\frac{1}{162} x 81 + x^{2} + \frac{1}{2} ln (\frac{x + 81 + x ^{2}}{9}))

= 81 (\frac{1}{162} x 81 + x^{2} + \frac{1}{2} ln (\frac{x + 81 + x ^{2}}{9}))

Add a constant to the solution

= 81 (\frac{1}{162} x 81 + x^{2} + \frac{1}{2} ln (\frac{x + 81 + x ^{2}}{9})) + C

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What is the integral of sqrt(81+x^2) ?
The integral of sqrt(81+x^2) is 81(1/162 xsqrt(81+x^2)+1/2 ln((|x+sqrt(81+x^2)|)/9))+C