What is the integral of sqrt(2x^2-8) ?

The integral of sqrt(2x^2-8) is 4sqrt(2)(1/8 xsqrt(x^2-4)-1/2 ln((|x+sqrt(x^2-4)|)/2))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(2x^2-8)

\int 2 x^{2} - 8 d x

42 (\frac{1}{8} x x^{2} - 4 - \frac{1}{2} ln (\frac{x + x ^{2} - 4}{2})) + C

Solution steps

\int 2 x^{2} - 8 d x

Apply Trigonometric Substitution

: \int 42 tan^{2} (u) sec (u) d u

= \int 42 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

= 42 \cdot \int tan^{2} (u) sec (u) d u

Rewrite using trig identities

= 42 \cdot \int (- 1 + sec^{2} (u)) sec (u) d u

Expand

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

= 42 \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

= 42 (- \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) ∣

= 42 (- 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{2}{2 2} x)

= 42 (- ln tan (arcsec (\frac{2}{2 2} x)) + sec (arcsec (\frac{2}{2 2} x)) + \frac{1}{2} sec (arcsec (\frac{2}{2 2} x)) tan (arcsec (\frac{2}{2 2} x)) + \frac{1}{2} ln tan (arcsec (\frac{2}{2 2} x)) + sec (arcsec (\frac{2}{2 2} x)))

Simplify

42 (- ln tan (arcsec (\frac{2}{2 2} x)) + sec (arcsec (\frac{2}{2 2} x)) + \frac{1}{2} sec (arcsec (\frac{2}{2 2} x)) tan (arcsec (\frac{2}{2 2} x)) + \frac{1}{2} ln tan (arcsec (\frac{2}{2 2} x)) + sec (arcsec (\frac{2}{2 2} x))) : 42 (\frac{1}{8} x x^{2} - 4 - \frac{1}{2} ln (\frac{x + x ^{2} - 4}{2}))

= 42 (\frac{1}{8} x x^{2} - 4 - \frac{1}{2} ln (\frac{x + x ^{2} - 4}{2}))

Add a constant to the solution

= 42 (\frac{1}{8} x x^{2} - 4 - \frac{1}{2} ln (\frac{x + x ^{2} - 4}{2})) + C

\int (x^{2} + 3 x) (sin (x^{2})) d x

\int \frac{x + \frac{1}{2}}{( x ^{2} - x + \frac{5}{4} )} d x

\int x 9 x^{2} - 1 d x

\int \frac{1}{a ^{2} + b ^{2} x ^{2}} d x

\int 18 x e^{3 x + 1} d x

What is the integral of sqrt(2x^2-8) ?
The integral of sqrt(2x^2-8) is 4sqrt(2)(1/8 xsqrt(x^2-4)-1/2 ln((|x+sqrt(x^2-4)|)/2))+C