What is the integral of 1/(x^2sqrt(x^2-5)) ?

The integral of 1/(x^2sqrt(x^2-5)) is (sqrt(x^2-5))/(5x)+C

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integral of 1/(x^2sqrt(x^2-5))

\int \frac{1}{x ^{2} x ^{2} - 5} d x

\frac{x ^{2} - 5}{5 x} + C

Solution steps

\int \frac{1}{x ^{2} x ^{2} - 5} d x

Apply Trigonometric Substitution

= \int \frac{1}{5 sec ( u )} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{5} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{5} sin (u)

Substitute back

u = arcsec (\frac{1}{5} x)

= \frac{1}{5} sin (arcsec (\frac{1}{5} x))

Simplify

\frac{1}{5} sin (arcsec (\frac{1}{5} x)) : \frac{x ^{2} - 5}{5 x}

= \frac{x ^{2} - 5}{5 x}

Add a constant to the solution

= \frac{x ^{2} - 5}{5 x} + C

y = x^{8 c o s (x)}

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

\int e^{3 x} sin (4 x) d x

x^{2} y^{^{''}} - 2 x y^{^{'}} + 2 y = 4 x^{2}

csc^{7} (2 t^{π} + π^{3})

What is the integral of 1/(x^2sqrt(x^2-5)) ?
The integral of 1/(x^2sqrt(x^2-5)) is (sqrt(x^2-5))/(5x)+C