Final answer:
To solve the integral of (1-sin³ x)/cos² x, we split it into two parts and integrate each separately. The first part integrates to tan(x) and for the second part, we use a substitution with u=sin x to find the antiderivative as -sin⁴(x)/4. Combining both gives us the answer: tan(x) - sin⁴(x)/4 + C.
Step-by-step explanation:
To solve the integral ∫(1-sin³ x)/cos² x dx, let's first simplify the integrand. We can rewrite the integral by splitting the fraction:
∫(1/cos² x) dx - ∫(sin³ x/cos² x) dx
The first term is straightforward to integrate as it is the secant squared function, whose antiderivative is the tangent function. So, we get:
∫(1/cos² x) dx = tan(x) + C
For the second term, we can use the substitution method. Let u = sin x, so that du = cos(x) dx. Then, our integral becomes:
∫(u³/cos² x) (cos(x) dx) = ∫(u³ du)
Which is a simple power function to integrate:
∫(u³ du) = ∛(u⁴)/4 + C = sin⁴(x)/4 + C
Combining both results, the final solution to the integral is:
tan(x) - sin⁴(x)/4 + C