Antiderivatives.What is the antiderivative of sin3x?
We have to find the anti derivative of sin 3x
Int [ sin 3x dx]
Let 3x = y => dx = dy / 3
=> Int [ (1/3)*sin y dy]
=> (1/3)*(-cos y)
=> (-1/3)* cos 3x + C
The anti derivative is (-1/3)* cos 3x + C
To find the antiderivative of sin 3x, we'll have to determine the indefinite integral of sin 3x.
Int sin 3x dx
We'll substitute 3x by t.
3x = t
We'll differentiate both sides:
3dx = dt
We'll divide by 3:
dx = dt/3
We'll re-write the integral of the function , changing the variable x to t:
Int sin t*(dt/3) = (1/3)*Int sin t dt
(1/3)*Int sin t dt = - cos t/3 +C
We'll substitute t by 3x:
Int sin 3x dx = - (cos 3x)/3 +C