Question

Найти первообразную функции : f (x) = 1/cos^2 x + sin x . пожалуйста.

Answer 1

f (x) = 1/cos^2 x + sin x
F(x) = tg x - cos x +C, где С=Сonst

Answer 2

Finding the Antiderivative of \( f(x) = \frac{1}{{\cos^2 x + \sin x}} \)

To find the antiderivative of the given function \( f(x) = \frac{1}{{\cos^2 x + \sin x}} \), we can use trigonometric identities and integration techniques.

First, we can rewrite the function using trigonometric identities. The identity \( \cos^2 x = 1 - \sin^2 x \) can be used to simplify the expression.

The function can be rewritten as: \[ f(x) = \frac{1}{{1 - \sin^2 x + \sin x}} \]

Now, we can use a substitution to simplify the expression further. Let's substitute \( u = \sin x \). Then, \( du = \cos x \, dx \).

After the substitution, the function becomes: \[ f(x) = \frac{1}{{1 - u^2 + u}} \, du \]

We can then use partial fraction decomposition and other integration techniques to find the antiderivative of the function.

I hope this helps! Let me know if you need further assistance.