What is the integral of (sqrt(12-x^2))/(x^2) ?

The integral of (sqrt(12-x^2))/(x^2) is -arcsin(1/(2sqrt(3))x)-(sqrt(12-x^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 (sqrt(12-x^2))/(x^2)

\int \frac{12 - x ^{2}}{x ^{2}} d x

- arcsin (\frac{1}{2 3} x) - \frac{12 - x ^{2}}{x} + C

Solution steps

\int \frac{12 - x ^{2}}{x ^{2}} d x

Apply Trigonometric Substitution

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

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

Rewrite using trig identities

= \int - 1 + csc^{2} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d u + \int csc^{2} (u) d u

\int 1 d u = u

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

= - u - cot (u)

Substitute back

u = arcsin (\frac{1}{2 3} x)

= - arcsin (\frac{1}{2 3} x) - cot (arcsin (\frac{1}{2 3} x))

Simplify

- arcsin (\frac{1}{2 3} x) - cot (arcsin (\frac{1}{2 3} x)) : - arcsin (\frac{1}{2 3} x) - \frac{12 - x ^{2}}{x}

= - arcsin (\frac{1}{2 3} x) - \frac{12 - x ^{2}}{x}

Add a constant to the solution

= - arcsin (\frac{1}{2 3} x) - \frac{12 - x ^{2}}{x} + C

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

\int (sin (x))^{2} \cdot cos (x) d x

\int cos^{1} (x) sin^{2} (x) d x

\int \frac{x ^{2} + 1}{( x ^{2} - 1 ) ( x + 1 )} d x

\int \frac{ln ^{4} ( x )}{x ( ln ^{2} ( x ) - 1 )} d x

What is the integral of (sqrt(12-x^2))/(x^2) ?
The integral of (sqrt(12-x^2))/(x^2) is -arcsin(1/(2sqrt(3))x)-(sqrt(12-x^2))/x+C