Math9999 said:Homework Statement
Find the integral of sin^7 x/(1+x^10) dx from -pi/2 to pi/2.
Homework Equations
None.
The Attempt at a Solution
sin^7 x means sinx to the 7th power. But how do I find this strange integral? I don't think u-substitution, trig identity, any of them will work.
Do you know what even and odd functions are?Math9999 said:I don't know anything about the integrand.
Math9999 said:I know that the sine functions are odd, right?
Math9999 said:That sin^7 (x) is also odd.
Math9999 said:An even function?
Exactly. And what do you get when you integrate an odd function from -a to a?Math9999 said:An odd function.
Math9999 said:0?
Math9999 said:0?
The power rule states that the integral of x^n is equal to (x^(n+1))/(n+1) + C, where C is a constant. To use the power rule, you must first rewrite the integral in the form of x^n. Then, simply plug in the value of n+1 and add the constant C.
The substitution method involves replacing a variable in the integral with a new variable, and then using the chain rule to solve for the new integral. This method is especially useful when the integral contains functions within functions, such as trigonometric functions.
Integration by parts is a method for finding integrals that involves breaking down the integral into two parts and using the product rule to solve for the integral. This method is particularly useful for integrals that contain products of functions.
Trigonometric identities can be used to rewrite trigonometric functions in terms of other trigonometric functions, which can make it easier to solve integrals. Common identities include sin^2(x) + cos^2(x) = 1 and tan(x) = sin(x)/cos(x).
A definite integral has specific upper and lower limits, while an indefinite integral does not. Indefinite integrals result in a function, while definite integrals result in a numerical value. Additionally, definite integrals are used to find the area under a curve, while indefinite integrals are used to find the antiderivative of a function.