What is the integral of (x^2+5)^{1/2} ?

The integral of (x^2+5)^{1/2} is 5(1/10 x(x^2+5)^{1/2}+1/2 ln| 1/(5^{1/2)}x+sqrt(1/5 (5+x^2))|)+C

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integral of (x^2+5)^{1/2}

\int (x^{2} + 5)^{\frac{1}{2}} d x

5 (\frac{1}{10} x (x^{2} + 5)^{\frac{1}{2}} + \frac{1}{2} ln \frac{1}{5 ^{\frac{1}{2}}} x + \frac{1}{5} (5 + x^{2})) + C

Solution steps

\int (x^{2} + 5)^{\frac{1}{2}} d x

Apply Trigonometric Substitution

: \int 5 sec^{3} (u) d u

= \int 5 sec^{3} (u) d u

Take the constant out:

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

= 5 \cdot \int sec^{3} (u) d u

Apply Integral Reduction

= 5 (\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) ∣

= 5 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arctan (\frac{1}{5 ^{\frac{1}{2}}} x)

= 5 \frac{sec ^{2} ( arctan ( \frac{1}{5 ^{\frac{1}{2}}} x ) ) sin ( arctan ( \frac{1}{5 ^{\frac{1}{2}}} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{5 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{5 ^{\frac{1}{2}}} x))

Simplify

5 \frac{sec ^{2} ( arctan ( \frac{1}{5 ^{\frac{1}{2}}} x ) ) sin ( arctan ( \frac{1}{5 ^{\frac{1}{2}}} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{5 ^{\frac{1}{2}}} x)) + sec (arctan (\frac{1}{5 ^{\frac{1}{2}}} x)) : 5 (\frac{1}{10} x (x^{2} + 5)^{\frac{1}{2}} + \frac{1}{2} ln \frac{1}{5 ^{\frac{1}{2}}} x + \frac{1}{5} (5 + x^{2}))

= 5 (\frac{1}{10} x (x^{2} + 5)^{\frac{1}{2}} + \frac{1}{2} ln \frac{1}{5 ^{\frac{1}{2}}} x + \frac{1}{5} (5 + x^{2}))

Add a constant to the solution

= 5 (\frac{1}{10} x (x^{2} + 5)^{\frac{1}{2}} + \frac{1}{2} ln \frac{1}{5 ^{\frac{1}{2}}} x + \frac{1}{5} (5 + x^{2})) + C

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What is the integral of (x^2+5)^{1/2} ?
The integral of (x^2+5)^{1/2} is 5(1/10 x(x^2+5)^{1/2}+1/2 ln| 1/(5^{1/2)}x+sqrt(1/5 (5+x^2))|)+C