That should be [itex]\displaystyle \int\frac{\cos^2(x)\sin^3(x)}{\cos^3(x)}\,dx[/itex]NWeid1 said:Homework Statement
∫cos(x)^2*tan(x)^3dx
Homework Equations
The Attempt at a Solution
Were learning Integration by parts and u substitution but this one I can't figure out. I tried making it ∫(cos(x))^2*(sin(x)^3)/(cos(x)^3)dx and then ∫tan(x)*sin(x)^2 but I don't know if I'm going in the right direction because I don't know how to solve from here.
The integral of cos(x)^2 * tan(x)^3dx is equal to (cos(x)^2 * tan(x)^2)/2 + ln|cos(x)| + C.
The purpose of using integration is to find the area under the curve represented by the function cos(x)^2 * tan(x)^3dx. This can be useful in solving real-world problems in fields such as physics and engineering.
The steps to solving this integral involve using integration techniques such as substitution and integration by parts. First, substitute u = cos(x) and du = -sin(x)dx. Then, use integration by parts with u = tan(x)^3 and dv = cos(x)dx. Finally, use the trigonometric identity tan(x)^2 = sec(x)^2 - 1 to simplify the integral.
Yes, this integral can also be solved using techniques such as trigonometric identities and partial fractions. However, integration is the most efficient and commonly used method for solving this type of integral.
Integrals have many applications in fields such as physics, engineering, and economics. They can be used to calculate distances, volumes, and areas, as well as solve problems involving rates of change and optimization.