Question

Помогите пожалуйста решить: Покажите что функция F (x) = +-cos x, является первообразной для функции f (x) = ++ sin x на всей числовой прямой

Answer 1

функция F (x) = e^(2x)+x^3-cosx определена и дифференцируема на всей
числовой оси. найдем производную F(x) используя табличные производные
(x^3)'=3x^2
(cosx)=-sinx
(e^(2x))'=2e^(2x)
(e^(2x)+x^3-cosx )'=2e^(2x)+3x^2+sinx=f(x)

Answer 2

Finding the Antiderivative of \( f(x) = \sin(x) \)

To show that the function \( F(x) = -\cos(x) \) is an antiderivative for the function \( f(x) = \sin(x) \) on the entire real number line, we can use the fundamental theorem of calculus. According to the theorem, if \( F(x) \) is an antiderivative of \( f(x) \), then the definite integral of \( f(x) \) from \( a \) to \( b \) is equal to \( F(b) - F(a) \).

Applying the Fundamental Theorem of Calculus

We can start by finding the derivative of \( F(x) \) and verifying that it matches the original function \( f(x) = \sin(x) \). The derivative of \( F(x) = -\cos(x) \) is:

\[ \frac{d}{dx}(-\cos(x)) = \sin(x) \]

This confirms that \( F(x) = -\cos(x) \) is an antiderivative of \( f(x) = \sin(x) \).

Conclusion

Therefore, the function \( F(x) = -\cos(x) \) is indeed a valid antiderivative for the function \( f(x) = \sin(x) \) on the entire real number line.