Answer:
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).
