How do I integrate arcsin (x) dx

INT arcsin (x) dx = x arcsin (x) + √ (1 - x^2) + C.

Step-by-step explanation:

Use substitution and integration by parts:

Let t = arc sinx then x = sin t and dx = cos t dt

So INT arcsin x dx = INT t cost dt

Now integrate by parts:-

let u = t and dv = cos t dt

then:

du = 1 and v = sin t dt

The formula for integation by parts is

INT u dv = uv - INT vdu so:

INT t cost dt = t sin t - INT 1 * sint dt

= t sint - ( - cos t) + C

= t sint + cos t + C.

Now substituting back for t, we have:

arcsin x * sin (arcsin x) + cos (arcsin x) + C.

Now sin (acrsin x) = x and cos (arcsin x) = √ (1 - x^2) so we have

INT arcsin x dx = x arcsin x + √ (1 - x^2) + C (answer).