What is the integral of arcsin(mx) ?

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

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integral of arcsin(mx)

\int arcsin (m x) d x

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

Solution steps

\int arcsin (m x) d x

Apply u-substitution

: \int \frac{arcsin ( u )}{m} d u

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = m x

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

Simplify

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

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

Add a constant to the solution

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

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