What is the integral of sqrt(t^2+5) ?

The integral of sqrt(t^2+5) is 5(1/10 tsqrt(5+t^2)+1/2 ln| 1/(sqrt(5))t+sqrt(1/5 (5+t^2))|)+C

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integral of sqrt(t^2+5)

\int t^{2} + 5 d t

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

Solution steps

\int t^{2} + 5 d t

Apply Trigonometric Substitution

= \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} t)

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

Simplify

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

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

Add a constant to the solution

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

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