∫((cos^3(x)/(1-sin^(2)) What is the derivative of that integral?
I have been trying to use trig identities but can't find one to simplify this equation. I can't find one for (cos^3(x) or (1-sin^(2))
My options
-sin(x) + C
sin(x) + C
(1/4)cos^(4)(x) + C
None of these
Is this supposed to be:
∫((cos^3(x)/(1-sin^2(x)) dx ??
1 - sin^2 x = cos^2 x
because cos^2 x + sin^2 x = 1
so you have
∫ cos(x) dx
the derivative of the integral is what you already have, cos x
the integral is sin x + c
Yes Thank you
Happy to help :)
To find the derivative of the integral ∫(cos^3x / (1-sin^2x)) dx, you will first need to simplify the integrand using trigonometric identities.
Let's start by simplifying the expression (cos^3x / (1-sin^2x)). Notice that cos^3x is the same as (cosx)^3, and sin^2x is the same as (sinx)^2. Replacing these terms in the integrand, we get:
(cosx)^3 / (1 - (sinx)^2).
Next, we can use the trigonometric identity 1 - (sinx)^2 = cos^2x, which is a special case of the Pythagorean identity. Substituting this identity into the integrand, we have:
(cosx)^3 / cos^2x.
Now, we can simplify further by canceling out one of the factors of cosx in the numerator and denominator:
cosx.
Therefore, the simplified integrand is just cosx.
Now that we have simplified the expression, let's find the derivative of the integral with respect to x. Since the original integral is ∫(cos^3x / (1-sin^2x)) dx, the derivative will be simply:
d/dx (∫cosx dx) = cosx + C.
So, the correct option is sin(x) + C.