What is the integral of 1/(sqrt(a^2-x^2)) ?

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

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integral of 1/(sqrt(a^2-x^2))

integral

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

arcsin (\frac{x}{a}) + C

Solution steps

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

= 1 \cdot arcsin (\frac{1}{a} x)

Simplify

1 \cdot arcsin (\frac{1}{a} x) : arcsin (\frac{x}{a})

= arcsin (\frac{x}{a})

Add a constant to the solution

= arcsin (\frac{x}{a}) + C

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

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

x^{2} (e^{- x})^{2}

1 - x

2 t^{2}

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