What is the integral of 1/(2x^2sqrt(x^2-16)) ?

The integral of 1/(2x^2sqrt(x^2-16)) is (sqrt(x^2-16))/(32x)+C

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integral of 1/(2x^2sqrt(x^2-16))

\int \frac{1}{2 x ^{2} x ^{2} - 16} d x

\frac{x ^{2} - 16}{32 x} + C

Solution steps

\int \frac{1}{2 x ^{2} x ^{2} - 16} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{x ^{2} x ^{2} - 16} d x

Apply Trigonometric Substitution

: \int \frac{1}{16 sec ( u )} d u

= \frac{1}{2} \cdot \int \frac{1}{16 sec ( u )} d u

Take the constant out:

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

= \frac{1}{2} \cdot \frac{1}{16} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{2} \cdot \frac{1}{16} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{2} \cdot \frac{1}{16} sin (u)

Substitute back

u = arcsec (\frac{1}{4} x)

= \frac{1}{2} \cdot \frac{1}{16} sin (arcsec (\frac{1}{4} x))

Simplify

\frac{1}{2} \cdot \frac{1}{16} sin (arcsec (\frac{1}{4} x)) : \frac{x ^{2} - 16}{32 x}

= \frac{x ^{2} - 16}{32 x}

Add a constant to the solution

= \frac{x ^{2} - 16}{32 x} + C

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What is the integral of 1/(2x^2sqrt(x^2-16)) ?
The integral of 1/(2x^2sqrt(x^2-16)) is (sqrt(x^2-16))/(32x)+C