What is the integral of arcsin(ax) ?

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

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integral of arcsin(ax)

\int arcsin (a x) d x

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

Solution steps

\int arcsin (a x) d x

Apply u-substitution

= \int \frac{arcsin ( u )}{a} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int arcsin (u) d u

Apply Integration By Parts

= \frac{1}{a} (u arcsin (u) - \int \frac{u}{1 - u ^{2}} d u)

\int \frac{u}{1 - u ^{2}} d u = - 1 - u^{2}

= \frac{1}{a} (u arcsin (u) - (- 1 - u^{2}))

Substitute back

u = a x

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

Simplify

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

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

Add a constant to the solution

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

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