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

1) Let's compute the integral:

\int \frac{1}{{\arcsin (x)}^{3} \cdot \sqrt{1 - x^{2}}} d x

For the math expression:

\frac{1}{{\arcsin (x)}^{3} \cdot \sqrt{1 - x^{2}}}

, since (a^n)' = n*a'*a^(n - 1), we try differentiating

{\arcsin (x)}^{- 3 + 1}

2) Let's perform the differentiation:

\frac{d}{d x} ({\arcsin (x)}^{- 3 + 1})

ⓘ The exponent is computed before the power:

- 3 + 1

-2

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

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

\arcsin (x)

ⓘ The derivative of x^n is n*x^(n-1):

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

-2 \cdot \frac{d}{d x} (\arcsin (x)) \cdot {\arcsin (x)}^{-3}

-2 \cdot \frac{d}{d x} (\arcsin (x)) \cdot {\arcsin (x)}^{-3}

ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

- 2 \cdot \frac{d}{d x} (\arcsin (x)) \cdot {\arcsin (x)}^{-3}

- \frac{2 \cdot \frac{d}{d x} (\arcsin (x))}{{\arcsin (x)}^{3}}

- \frac{2 \cdot \frac{d}{d x} (\arcsin (x))}{{\arcsin (x)}^{3}}

ⓘ The derivative of arcsin(x) is 1/sqrt(1 - x^2):

\frac{d}{d x} (\arcsin (x))

\frac{1}{\sqrt{1 - x^{2}}}

- \frac{2 \cdot \frac{1}{\sqrt{1 - x^{2}}}}{{\arcsin (x)}^{3}}

ⓘ Multiplying a fraction by a number multiplies its numerator:

2 \cdot \frac{1}{\sqrt{1 - x^{2}}}

\frac{2 \cdot 1}{\sqrt{1 - x^{2}}}

\frac{2}{\sqrt{1 - x^{2}}}

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

ⓘ Dividing a fraction by

{\arcsin (x)}^{3}

is multiplying its denominator by

{\arcsin (x)}^{3}

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

- \frac{2}{\sqrt{1 - x^{2}} \cdot {\arcsin (x)}^{3}}

- \frac{2}{\sqrt{1 - x^{2}} \cdot {\arcsin (x)}^{3}}

The computed derivative matches the original math expression up to a constant factor. Therefore, the integral is:

ⓘ The derivative of

- \frac{1}{2} \cdot \frac{1}{{\arcsin (x)}^{2}}

\frac{1}{{\arcsin (x)}^{3} \cdot \sqrt{1 - x^{2}}}

, so

- \frac{1}{2} \cdot \frac{1}{{\arcsin (x)}^{2}}

is an antiderivative of

\frac{1}{{\arcsin (x)}^{3} \cdot \sqrt{1 - x^{2}}}

\int \frac{1}{{\arcsin (x)}^{3} \cdot \sqrt{1 - x^{2}}} d x

- \frac{1}{2} \cdot \frac{1}{{\arcsin (x)}^{2}}

- \frac{1}{2} \cdot \frac{1}{{\arcsin (x)}^{2}}

ⓘ To multiply fractions, multiply the numerators together and the denominators together:

- \frac{1}{2} \cdot \frac{1}{{\arcsin (x)}^{2}}

- \frac{1 \cdot 1}{2 \cdot {\arcsin (x)}^{2}}

- \frac{1}{2 \cdot {\arcsin (x)}^{2}}

- \frac{1}{2 \cdot {\arcsin (x)}^{2}}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

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