What is the integral of 1/(w^2sqrt(w^2-7)) ?

The integral of 1/(w^2sqrt(w^2-7)) is (sqrt(w^2-7))/(7w)+C

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integral of 1/(w^2sqrt(w^2-7))

\int \frac{1}{w ^{2} w ^{2} - 7} d w

\frac{w ^{2} - 7}{7 w} + C

Solution steps

\int \frac{1}{w ^{2} w ^{2} - 7} d w

Apply Trigonometric Substitution

: \int \frac{1}{7 sec ( u )} d u

= \int \frac{1}{7 sec ( u )} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{7} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{1}{7} sin (u)

Substitute back

u = arcsec (\frac{1}{7} w)

= \frac{1}{7} sin (arcsec (\frac{1}{7} w))

Simplify

\frac{1}{7} sin (arcsec (\frac{1}{7} w)) : \frac{w ^{2} - 7}{7 w}

= \frac{w ^{2} - 7}{7 w}

Add a constant to the solution

= \frac{w ^{2} - 7}{7 w} + C

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What is the integral of 1/(w^2sqrt(w^2-7)) ?
The integral of 1/(w^2sqrt(w^2-7)) is (sqrt(w^2-7))/(7w)+C