What is the integral of 1/((1-x^2)^{1/2)} ?

The integral of 1/((1-x^2)^{1/2)} is arcsin(x)+C

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integral of 1/((1-x^2)^{1/2)}

integral

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

arcsin (x) + C

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

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

Apply Trigonometric Substitution

: \int 1 d u

= \int 1 d u

Integral of a constant:

\int a d x = a x

= 1 \cdot u

Substitute back

u = arcsin (x)

= 1 \cdot arcsin (x)

Simplify

= arcsin (x)

Add a constant to the solution

= arcsin (x) + C

\frac{1}{( x ^{2} + 1 ) ^{\frac{3}{2}}}

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

\frac{y}{( x ^{2} + y ^{2} )}

sin^{5} (x) cos^{3} (x)

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

What is the integral of 1/((1-x^2)^{1/2)} ?
The integral of 1/((1-x^2)^{1/2)} is arcsin(x)+C