What is the integral of 1/(x^2sqrt(81x^2-49)) ?

The integral of 1/(x^2sqrt(81x^2-49)) is (sqrt(81x^2-49))/(49x)+C

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integral of 1/(x^2sqrt(81x^2-49))

\int \frac{1}{x ^{2} 81 x ^{2} - 49} d x

\frac{81 x ^{2} - 49}{49 x} + C

Solution steps

\int \frac{1}{x ^{2} 81 x ^{2} - 49} d x

Apply Trigonometric Substitution

: \int \frac{9}{49 sec ( u )} d u

= \int \frac{9}{49 sec ( u )} d u

Take the constant out:

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

= \frac{9}{49} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{9}{49} \cdot \int cos (u) d u

Use the common integral:

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

= \frac{9}{49} sin (u)

Substitute back

u = arcsec (\frac{9}{7} x)

= \frac{9}{49} sin (arcsec (\frac{9}{7} x))

Simplify

\frac{9}{49} sin (arcsec (\frac{9}{7} x)) : \frac{81 x ^{2} - 49}{49 x}

= \frac{81 x ^{2} - 49}{49 x}

Add a constant to the solution

= \frac{81 x ^{2} - 49}{49 x} + C

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\frac{d}{d x} (x^{2} sin^{7} (x) + x cos^{- 2} (x))

f (x) = x^{2} + 7

What is the integral of 1/(x^2sqrt(81x^2-49)) ?
The integral of 1/(x^2sqrt(81x^2-49)) is (sqrt(81x^2-49))/(49x)+C