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

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

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integral of 1/(x^2sqrt(2x^2+32))

\int \frac{1}{x ^{2} 2 x ^{2} + 32} d x

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

Solution steps

\int \frac{1}{x ^{2} 2 x ^{2} + 32} d x

\frac{1}{x ^{2} 2 x ^{2} + 32} = \frac{1}{2} \frac{1}{x ^{2} x ^{2} + 16}

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

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{1}{v}

= \frac{1}{2} \cdot \frac{1}{16} (- \frac{1}{v})

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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