sugarxsweet said:Homework Statement
Given that n is a positive integer, prove ∫sin^n(x)dx=∫cos^n(x)dx from 0 -> pi/2
Homework Equations
Perhaps sin^2(x)+cos^2(x)=1? Not sure.
The Attempt at a Solution
I honestly don't even know where to start. Should I set u=sin(x) or cos(x)? Doesn't seem to get the right answer either way...
Use the substitution, [itex]u=\frac{\pi}{2}-x[/itex] to show that [itex]\displaystyle \int_{x=0}^{x=\pi/2}\sin^n(\frac{\pi}{2}-x)\,dx = \int_{u=\pi/2}^{u=0}-\sin(u)\,du\ .[/itex]sugarxsweet said:Thanks! I got so far:
∫cos^n(x)dx from 0 to pi/2
=∫sin^n(pi/2-x)dx from 0 to pi/2
=∫sin^n(-x)dx from -pi/2 to 0
=∫sin^n(x)dx from 0 to pi/2
Does this look right to you? Thanks for the hint!
The Substitution Rule with sin^n(x) is a method used in integration to evaluate integrals involving powers of sine. It involves substituting a new variable, u, for the sine function, which allows for the integral to be simplified and evaluated using basic integration techniques.
To use the Substitution Rule with sin^n(x), you first identify the integral as one that can be solved using this method. Then, you choose a new variable, u, to substitute for the sine function. This variable should be chosen such that when u is substituted back into the integral, it simplifies and becomes easier to evaluate. After substituting, you can use basic integration techniques to solve the integral in terms of u, and then substitute back in for the original variable, x.
The Substitution Rule with sin^n(x) allows for the evaluation of integrals that would otherwise be difficult or impossible to solve using basic integration techniques. It simplifies the integral and makes it more manageable, often resulting in a solution that can be easily integrated using methods such as the power rule or u-substitution.
While the Substitution Rule with sin^n(x) is a useful method for evaluating integrals, it does have its limitations. It can only be used for integrals involving powers of sine, and it may not always result in a simplified integral. Additionally, it can sometimes be difficult to choose the right substitution variable, which can complicate the integration process.
The best way to practice using the Substitution Rule with sin^n(x) is to solve a variety of integrals using this method. You can find practice problems online or in a calculus textbook. Additionally, working with a tutor or attending a study group can help you gain a better understanding of the Substitution Rule and how to apply it to different types of integrals involving sine.