- #1
mugzieee
- 77
- 0
i don't understand how the antiderviative of 2 sin (2x) could be -cos(2x), what happens to the 2?
also how can the antiderivative of 5e^(5t) be e^(5t)?
also how can the antiderivative of 5e^(5t) be e^(5t)?
mugzieee said:i don't understand how the antiderviative of 2 sin (2x) could be -cos(2x), what happens to the 2?
also how can the antiderivative of 5e^(5t) be e^(5t)?
The antiderviative (or integral) of 2 sin(2x) is -cos(2x) + C, where C is a constant. This can be found by using the power rule for integrals and the chain rule to account for the 2x inside the sine function.
The negative sign in front of the cosine function in the antiderviative is due to the negative sign in front of the sin(2x) in the original function. This is a result of the chain rule, where the derivative of cos(2x) is -sin(2x).
No, the antiderviative of 2 sin(2x) = -cos(2x) + C is the simplest form. It cannot be simplified any further.
This can be verified by taking the derivative of -cos(2x) and using the chain rule, which will result in 2 sin(2x). Therefore, -cos(2x) is the antiderviative of 2 sin(2x).
Yes, the antiderviative of 2 sin(2x) can also be written as -1/2 sin(2x) + C or cos(2x + π/2) + C. These are all equivalent forms of the antiderviative.