What is the integral of (sqrt(x^2-a^2))/(x^4) ?

The integral of (sqrt(x^2-a^2))/(x^4) is ((x^2-a^2)^{3/2})/(3a^2x^3)+C

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integral of (sqrt(x^2-a^2))/(x^4)

\int \frac{x ^{2} - a ^{2}}{x ^{4}} d x

\frac{( x ^{2} - a ^{2} ) ^{\frac{3}{2}}}{3 a ^{2} x ^{3}} + C

Solution steps

\int \frac{x ^{2} - a ^{2}}{x ^{4}} d x

Apply Trigonometric Substitution

: \int \frac{tan ^{2} ( u )}{a ^{2} sec ^{3} ( u )} d u

= \int \frac{tan ^{2} ( u )}{a ^{2} sec ^{3} ( u )} d u

Take the constant out:

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

= \frac{1}{a ^{2}} \cdot \int \frac{tan ^{2} ( u )}{sec ^{3} ( u )} d u

Rewrite using trig identities

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

Apply u-substitution

: \int v^{2} d v

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

Apply the Power Rule

: \frac{v ^{3}}{3}

= \frac{1}{a ^{2}} \cdot \frac{v ^{3}}{3}

Substitute back

= \frac{1}{a ^{2}} \cdot \frac{sin ^{3} ( arcsec ( \frac{1}{a} x ) )}{3}

Simplify

\frac{1}{a ^{2}} \cdot \frac{sin ^{3} ( arcsec ( \frac{1}{a} x ) )}{3} : \frac{( x ^{2} - a ^{2} ) ^{\frac{3}{2}}}{3 a ^{2} x ^{3}}

= \frac{( x ^{2} - a ^{2} ) ^{\frac{3}{2}}}{3 a ^{2} x ^{3}}

Add a constant to the solution

= \frac{( x ^{2} - a ^{2} ) ^{\frac{3}{2}}}{3 a ^{2} x ^{3}} + C

(2 x)^{^{'}}

\int \frac{( cos ^{3} ( x ) )}{3} d x

x \to 3 lim ((2 x^{2} - 4 x - 8)^{6})

\frac{\partial}{\partial x} (\frac{4 x}{y})

\int \frac{4 x - 2}{x ^{3} - x ^{2} - 2 x} d x

What is the integral of (sqrt(x^2-a^2))/(x^4) ?
The integral of (sqrt(x^2-a^2))/(x^4) is ((x^2-a^2)^{3/2})/(3a^2x^3)+C