togame said:Homework Statement
I just need to find the anti-derivative of this equation:
[itex]\int x(sin^2(3x)cos(3x))dx[/itex]
Homework Equations
[itex]sin^2(x)+cos^2(x)=1[/itex]
The Attempt at a Solution
I'm not really sure where to start here. I tried to do a substitution first but couldn't make it work. So then I tried to put sin^2 in terms of cos^2 but that didn't seem to help me out any more. Anyone have an idea of how to tackle this problem? Thanks in advance!
Curious3141 said:Integrate by parts, with u = x, dv = [itex]\sin^2(3x)\cos(3x)dx[/itex].
To integrate [itex]\sin^2(3x)\cos(3x)dx[/itex], use the substitution [itex]y = \sin(3x)[/itex]
An anti-derivative of a trigonometric equation is a function whose derivative is equal to the original trigonometric equation. It is also known as an indefinite integral.
To find the anti-derivative of a trigonometric equation, you can use the basic integration rules for trigonometric functions, such as the power rule, substitution, and integration by parts.
The most common trigonometric functions used in anti-derivatives are sine, cosine, tangent, cotangent, secant, and cosecant. These functions are used in various combinations and forms to find the anti-derivative of a trigonometric equation.
The anti-derivative of a trigonometric equation is important because it helps in solving many real-world problems that involve trigonometric functions. It is also a fundamental concept in calculus and is used extensively in higher-level mathematics and science courses.
No, not every trigonometric equation has an anti-derivative. Some trigonometric equations are not integrable, meaning they do not have a closed-form solution. In such cases, numerical methods or approximation techniques are used to find the anti-derivative.