What is the integral of sqrt(x^2-10) ?

The integral of sqrt(x^2-10) is 10(1/20 xsqrt(x^2-10)-1/2 ln| 1/(sqrt(10))x+sqrt(1/10 (x^2-10))|)+C

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integral of sqrt(x^2-10)

\int x^{2} - 10 d x

10 (\frac{1}{20} x x^{2} - 10 - \frac{1}{2} ln \frac{1}{10} x + \frac{1}{10} (x^{2} - 10)) + C

Solution steps

\int x^{2} - 10 d x

Apply Trigonometric Substitution

: \int 10 tan^{2} (u) sec (u) d u

= \int 10 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

= 10 \cdot \int tan^{2} (u) sec (u) d u

Rewrite using trig identities

= 10 \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

Expand

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

= 10 \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

= 10 (- \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) ∣

= 10 (- 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{1}{10} x)

= 10 (- ln tan (arcsec (\frac{1}{10} x)) + sec (arcsec (\frac{1}{10} x)) + \frac{1}{2} sec (arcsec (\frac{1}{10} x)) tan (arcsec (\frac{1}{10} x)) + \frac{1}{2} ln tan (arcsec (\frac{1}{10} x)) + sec (arcsec (\frac{1}{10} x)))

Simplify

10 (- ln tan (arcsec (\frac{1}{10} x)) + sec (arcsec (\frac{1}{10} x)) + \frac{1}{2} sec (arcsec (\frac{1}{10} x)) tan (arcsec (\frac{1}{10} x)) + \frac{1}{2} ln tan (arcsec (\frac{1}{10} x)) + sec (arcsec (\frac{1}{10} x))) : 10 (\frac{1}{20} x x^{2} - 10 - \frac{1}{2} ln \frac{1}{10} x + \frac{1}{10} (x^{2} - 10))

= 10 (\frac{1}{20} x x^{2} - 10 - \frac{1}{2} ln \frac{1}{10} x + \frac{1}{10} (x^{2} - 10))

Add a constant to the solution

= 10 (\frac{1}{20} x x^{2} - 10 - \frac{1}{2} ln \frac{1}{10} x + \frac{1}{10} (x^{2} - 10)) + C

x \to 0 - lim (\frac{( 4 - x ) - 0}{4})

\frac{d}{d x} (2 e^{2 x} x)

\int e^{3 x + 1} d x

\int_{2}^{6} \frac{1}{x ^{2} - 1} d x

\frac{\partial}{\partial y} (e^{y^{2}})

What is the integral of sqrt(x^2-10) ?
The integral of sqrt(x^2-10) is 10(1/20 xsqrt(x^2-10)-1/2 ln| 1/(sqrt(10))x+sqrt(1/10 (x^2-10))|)+C