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

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

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

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

- \frac{1 + x ^{2}}{x} + C

Solution steps

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

Apply Trigonometric Substitution

: \int cot^{2} (u) sec (u) d u

= \int cot^{2} (u) sec (u) d u

Simplify

cot^{2} (u) sec (u) : cot (u) csc (u)

= \int cot (u) csc (u) d u

Apply u-substitution

: \int - 1 d v

= \int - 1 d v

Integral of a constant:

\int a d x = a x

= (- 1) v

Substitute back

= (- 1) csc (arctan (x))

Simplify

(- 1) csc (arctan (x)) : - \frac{1 + x ^{2}}{x}

= - \frac{1 + x ^{2}}{x}

Add a constant to the solution

= - \frac{1 + x ^{2}}{x} + C

\int \frac{8}{9 + r ^{2}} d r

f (x) = \frac{( 4 x ) ^{3}}{( 3 - 5 x ) ^{5}}

\int 4 x ln (4 x) d x

ln (e^{x} x)

\frac{1}{1 - x}, (\frac{π}{4})

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