What is the integral of 1/(x^2sqrt(9x^2+1)) ?

The integral of 1/(x^2sqrt(9x^2+1)) is -(sqrt(1+9x^2))/x+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(x^2sqrt(9x^2+1))

\int \frac{1}{x ^{2} 9 x ^{2} + 1} d x

- \frac{1 + 9 x ^{2}}{x} + C

Solution steps

\int \frac{1}{x ^{2} 9 x ^{2} + 1} d x

Apply Trigonometric Substitution

: \int 3 cot^{2} (u) sec (u) d u

= \int 3 cot^{2} (u) sec (u) d u

Take the constant out:

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

= 3 \cdot \int cot^{2} (u) sec (u) d u

Simplify

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

= 3 \cdot \int cot (u) csc (u) d u

Apply u-substitution

: \int - 1 d v

= 3 \cdot \int - 1 d v

Integral of a constant:

\int a d x = a x

= 3 (- 1) v

Substitute back

= 3 (- 1) csc (arctan (3 x))

Simplify

3 (- 1) csc (arctan (3 x)) : - \frac{1 + 9 x ^{2}}{x}

= - \frac{1 + 9 x ^{2}}{x}

Add a constant to the solution

= - \frac{1 + 9 x ^{2}}{x} + C

\int x sin (3 x - 1) d x

\int \frac{tan ( x )}{ln ( cos ( x ) )} d x

\int (4 x + 3) (2 x^{2} + 3 x) 3 d x

\int cos (3 x) sin (2 x) d x

What is the integral of 1/(x^2sqrt(9x^2+1)) ?
The integral of 1/(x^2sqrt(9x^2+1)) is -(sqrt(1+9x^2))/x+C