# What is the integral of #int sin^3x/cos^2x dx#?

I found:

By signing up, you agree to our Terms of Service and Privacy Policy

To solve the integral ∫(sin^3(x) / cos^2(x)) dx, we can use the substitution method. Let u = sin(x) and du = cos(x) dx. Then the integral becomes ∫(u^3 / cos(x) du). Using trigonometric identity cos^2(x) = 1 - sin^2(x), we can replace cos(x) with √(1 - u^2). The integral then becomes ∫(u^3 / √(1 - u^2)) du. This integral can be solved using a trigonometric substitution. Let u = sin(θ), then du = cos(θ) dθ and √(1 - u^2) = cos(θ). Substituting these into the integral gives us ∫(sin^3(θ)) dθ. Using trigonometric identity sin^3(θ) = (3sin(θ) - sin(3θ))/4, the integral becomes ∫((3sin(θ) - sin(3θ))/4) dθ. Integrating term by term yields (-3cos(θ) + cos(3θ)/12 + C), where C is the constant of integration. Finally, substitute back u = sin(x) and θ = arcsin(u) to get the integral in terms of x: (-3cos(arcsin(u)) + cos(3arcsin(u))/12 + C). Simplifying further, we get (-3√(1 - u^2) + u(9 - 12u^2)/12 + C), which is the integral of the given function.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the definite integral for: # dx / (x sqrt(lnx))# for the intervals #[1, e^8]#?
- What is the integral of #int 1/(x^2+4)dx#?
- How do you find the sum of the infinite geometric series #Sigma 2(-2/3)^n# from n=0 to #oo#?
- How do you find the antiderivative of #4/sqrtx#?
- How do you find a formula for the sum n terms #Sigma(2i)/n(2/n)# and then find the limit as #n->oo#?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7