What is the integral of (x^2)/(sqrt(x^2-49)) ?

The integral of (x^2)/(sqrt(x^2-49)) is 49(1/98 xsqrt(x^2-49)+1/2 ln((|sqrt(x^2-49)+x|)/7))+C

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integral of (x^2)/(sqrt(x^2-49))

\int \frac{x ^{2}}{x ^{2} - 49} d x

49 (\frac{1}{98} x x^{2} - 49 + \frac{1}{2} ln (\frac{x ^{2} - 49 + x}{7})) + C

Solution steps

\int \frac{x ^{2}}{x ^{2} - 49} d x

Apply Trigonometric Substitution

: \int 49 sec^{3} (u) d u

= \int 49 sec^{3} (u) d u

Take the constant out:

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

= 49 \cdot \int sec^{3} (u) d u

Apply Integral Reduction

= 49 (\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) ∣

= 49 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arcsec (\frac{1}{7} x)

= 49 (\frac{sec ^{2} ( arcsec ( \frac{1}{7} x ) ) sin ( arcsec ( \frac{1}{7} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{1}{7} x)) + sec (arcsec (\frac{1}{7} x)))

Simplify

49 (\frac{sec ^{2} ( arcsec ( \frac{1}{7} x ) ) sin ( arcsec ( \frac{1}{7} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{1}{7} x)) + sec (arcsec (\frac{1}{7} x))) : 49 (\frac{1}{98} x x^{2} - 49 + \frac{1}{2} ln (\frac{x ^{2} - 49 + x}{7}))

= 49 (\frac{1}{98} x x^{2} - 49 + \frac{1}{2} ln (\frac{x ^{2} - 49 + x}{7}))

Add a constant to the solution

= 49 (\frac{1}{98} x x^{2} - 49 + \frac{1}{2} ln (\frac{x ^{2} - 49 + x}{7})) + C

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What is the integral of (x^2)/(sqrt(x^2-49)) ?
The integral of (x^2)/(sqrt(x^2-49)) is 49(1/98 xsqrt(x^2-49)+1/2 ln((|sqrt(x^2-49)+x|)/7))+C