What is the integral of (x^2)/(sqrt(x^2-36)) ?

The integral of (x^2)/(sqrt(x^2-36)) is 36(1/72 xsqrt(x^2-36)+1/2 ln((|sqrt(x^2-36)+x|)/6))+C

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integral of (x^2)/(sqrt(x^2-36))

\int \frac{x ^{2}}{x ^{2} - 36} d x

36 (\frac{1}{72} x x^{2} - 36 + \frac{1}{2} ln (\frac{x ^{2} - 36 + x}{6})) + C

Solution steps

\int \frac{x ^{2}}{x ^{2} - 36} d x

Apply Trigonometric Substitution

: \int 36 sec^{3} (u) d u

= \int 36 sec^{3} (u) d u

Take the constant out:

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

= 36 \cdot \int sec^{3} (u) d u

Apply Integral Reduction

= 36 (\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) ∣

= 36 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arcsec (\frac{1}{6} x)

= 36 (\frac{sec ^{2} ( arcsec ( \frac{1}{6} x ) ) sin ( arcsec ( \frac{1}{6} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{1}{6} x)) + sec (arcsec (\frac{1}{6} x)))

Simplify

36 (\frac{sec ^{2} ( arcsec ( \frac{1}{6} x ) ) sin ( arcsec ( \frac{1}{6} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{1}{6} x)) + sec (arcsec (\frac{1}{6} x))) : 36 (\frac{1}{72} x x^{2} - 36 + \frac{1}{2} ln (\frac{x ^{2} - 36 + x}{6}))

= 36 (\frac{1}{72} x x^{2} - 36 + \frac{1}{2} ln (\frac{x ^{2} - 36 + x}{6}))

Add a constant to the solution

= 36 (\frac{1}{72} x x^{2} - 36 + \frac{1}{2} ln (\frac{x ^{2} - 36 + x}{6})) + C

x \to 2 lim (3 x + h^{2} - 7)

x \to 1 lim (\frac{2 x - 3}{x + 5})

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

\frac{d y}{d t} + 3 y = 13 sin (2 t)

\int \frac{2 x + 1}{x ( x + 2 ) ^{3}} d x

What is the integral of (x^2)/(sqrt(x^2-36)) ?
The integral of (x^2)/(sqrt(x^2-36)) is 36(1/72 xsqrt(x^2-36)+1/2 ln((|sqrt(x^2-36)+x|)/6))+C