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

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

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

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

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

Solution steps

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

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} x)

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

Simplify

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

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

Add a constant to the solution

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

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

\int_{0}^{1} \frac{2}{( x + 1 ) ( x ^{2} + 1 )} d x

\frac{d}{d x} (x^{3} + 3 x^{2} - 24 x - 80)

f (x) = x^{3}, at x = 0

f (x) = \frac{3 x ^{2} - 18 x + 15}{5}

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