What is the integral of 1/(y^3sqrt(y^2-9)) ?

The integral of 1/(y^3sqrt(y^2-9)) is 1/54 (arcsec(1/3 y)+1/2 sin(2arcsec(1/3 y)))+C

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integral of 1/(y^3sqrt(y^2-9))

\int \frac{1}{y ^{3} y ^{2} - 9} d y

\frac{1}{54} (arcsec (\frac{1}{3} y) + \frac{1}{2} sin (2 arcsec (\frac{1}{3} y))) + C

Solution steps

\int \frac{1}{y ^{3} y ^{2} - 9} d y

Apply Trigonometric Substitution

: \int \frac{1}{27 sec ^{2} ( u )} d u

= \int \frac{1}{27 sec ^{2} ( u )} d u

Take the constant out:

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

= \frac{1}{27} \cdot \int \frac{1}{sec ^{2} ( u )} d u

Rewrite using trig identities

= \frac{1}{27} \cdot \int \frac{1 + cos ( 2 u )}{2} d u

Take the constant out:

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

= \frac{1}{27} \cdot \frac{1}{2} \cdot \int 1 + cos (2 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{1}{27} \cdot \frac{1}{2} (\int 1 d u + \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= \frac{1}{27} \cdot \frac{1}{2} (u + \frac{1}{2} sin (2 u))

Substitute back

u = arcsec (\frac{1}{3} y)

= \frac{1}{27} \cdot \frac{1}{2} (arcsec (\frac{1}{3} y) + \frac{1}{2} sin (2 arcsec (\frac{1}{3} y)))

Simplify

\frac{1}{27} \cdot \frac{1}{2} (arcsec (\frac{1}{3} y) + \frac{1}{2} sin (2 arcsec (\frac{1}{3} y))) : \frac{1}{54} (arcsec (\frac{1}{3} y) + \frac{1}{2} sin (2 arcsec (\frac{1}{3} y)))

= \frac{1}{54} (arcsec (\frac{1}{3} y) + \frac{1}{2} sin (2 arcsec (\frac{1}{3} y)))

Add a constant to the solution

= \frac{1}{54} (arcsec (\frac{1}{3} y) + \frac{1}{2} sin (2 arcsec (\frac{1}{3} y))) + C

\frac{d}{d θ} (sin (4 θ))

\int x (x^{2} - 2) d x

(f (t) \circ g (t))^{'}

\int_{2}^{3} \frac{x + 1}{x ^{2} - x} d x

x \to 1 lim (\frac{3 x + 2}{4 x - 5})

What is the integral of 1/(y^3sqrt(y^2-9)) ?
The integral of 1/(y^3sqrt(y^2-9)) is 1/54 (arcsec(1/3 y)+1/2 sin(2arcsec(1/3 y)))+C