What is the integral of sqrt(25x^2-4) ?

The integral of sqrt(25x^2-4) is 1/10 (5xsqrt(25x^2-4)-4ln((|5x+sqrt(25x^2-4)|)/2))+C

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integral of sqrt(25x^2-4)

\int 25 x^{2} - 4 d x

\frac{1}{10} (5 x 25 x^{2} - 4 - 4 ln (\frac{5 x + 25 x ^{2} - 4}{2})) + C

Solution steps

\int 25 x^{2} - 4 d x

Apply Trigonometric Substitution

: \int \frac{4}{5} tan^{2} (u) sec (u) d u

= \int \frac{4}{5} tan^{2} (u) sec (u) d u

Take the constant out:

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

= \frac{4}{5} \cdot \int tan^{2} (u) sec (u) d u

Rewrite using trig identities

= \frac{4}{5} \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

Expand

(- 1 + sec^{2} (u)) sec (u) : - sec (u) + sec^{3} (u)

= \frac{4}{5} \cdot \int - sec (u) + sec^{3} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{4}{5} (- \int sec (u) d u + \int sec^{3} (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

\int sec^{3} (u) d u = \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

= \frac{4}{5} (- ln ∣ tan (u) + sec (u) ∣ + \frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arcsec (\frac{5}{2} x)

= \frac{4}{5} (- ln tan (arcsec (\frac{5}{2} x)) + sec (arcsec (\frac{5}{2} x)) + \frac{1}{2} sec (arcsec (\frac{5}{2} x)) tan (arcsec (\frac{5}{2} x)) + \frac{1}{2} ln tan (arcsec (\frac{5}{2} x)) + sec (arcsec (\frac{5}{2} x)))

Simplify

\frac{4}{5} (- ln tan (arcsec (\frac{5}{2} x)) + sec (arcsec (\frac{5}{2} x)) + \frac{1}{2} sec (arcsec (\frac{5}{2} x)) tan (arcsec (\frac{5}{2} x)) + \frac{1}{2} ln tan (arcsec (\frac{5}{2} x)) + sec (arcsec (\frac{5}{2} x))) : \frac{1}{10} (5 x 25 x^{2} - 4 - 4 ln (\frac{5 x + 25 x ^{2} - 4}{2}))

= \frac{1}{10} (5 x 25 x^{2} - 4 - 4 ln (\frac{5 x + 25 x ^{2} - 4}{2}))

Add a constant to the solution

= \frac{1}{10} (5 x 25 x^{2} - 4 - 4 ln (\frac{5 x + 25 x ^{2} - 4}{2})) + C

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f (x) = x^{2} - 4, at 5

What is the integral of sqrt(25x^2-4) ?
The integral of sqrt(25x^2-4) is 1/10 (5xsqrt(25x^2-4)-4ln((|5x+sqrt(25x^2-4)|)/2))+C