What is the integral of 1/((x^2-121)^{3/2)} ?

The integral of 1/((x^2-121)^{3/2)} is -x/(121(x^2-121)^{1/2)}+C

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integral of 1/((x^2-121)^{3/2)}

\int \frac{1}{( x ^{2} - 121 ) ^{\frac{3}{2}}} d x

- \frac{x}{121 ( x ^{2} - 121 ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{1}{( x ^{2} - 121 ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

: \int \frac{sec ( u )}{121 tan ^{2} ( u )} d u

= \int \frac{sec ( u )}{121 tan ^{2} ( u )} d u

Take the constant out:

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

= \frac{1}{121} \cdot \int \frac{sec ( u )}{tan ^{2} ( u )} d u

Rewrite using trig identities

= \frac{1}{121} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

Apply u-substitution

: \int \frac{1}{v ^{2}} d v

= \frac{1}{121} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= \frac{1}{121} (- \frac{1}{v})

Substitute back

= \frac{1}{121} (- \frac{1}{sin ( arcsec ( \frac{1}{11} x ) )})

Simplify

\frac{1}{121} (- \frac{1}{sin ( arcsec ( \frac{1}{11} x ) )}) : - \frac{x}{121 ( x ^{2} - 121 ) ^{\frac{1}{2}}}

= - \frac{x}{121 ( x ^{2} - 121 ) ^{\frac{1}{2}}}

Add a constant to the solution

= - \frac{x}{121 ( x ^{2} - 121 ) ^{\frac{1}{2}}} + C

8 x^{6} + 5 w^{2} + 7 w

y = 3 x^{2} + 10 x + 2, at x = 1

x \to \infty lim (2^{2 x})

y = \frac{3 x - 1}{6 x + 5}

(\frac{1}{2} arcsin (x) + \frac{1}{4} sin (2 arcsin (x)))^{^{'}}

What is the integral of 1/((x^2-121)^{3/2)} ?
The integral of 1/((x^2-121)^{3/2)} is -x/(121(x^2-121)^{1/2)}+C