What voko said is exactly right. To elaborate, these two apparently different integrals differ only by a constant.Jude075 said:If you integrate it using substitution, you get -cos2(x)/2but if you use double angle formula to rewrite the problem, it will be ∫1/2 sin(2x), and integrate it, you get -cos(2x)/4. isn't it weird?
To integrate sin2x*cos2x using double angle, we can use the identity sin2x = 2sinx*cosx. This gives us the integral of 2sinx*cosx*cos2x, which can be simplified using the double angle identity cos2x = 2cos^2x - 1. We then have the integral of 2sinx*cosx*(2cos^2x - 1), which can be further simplified to 4cos^2x*sinx - 2sinx. Using the power rule and the chain rule, we can then find the final answer.
Yes, the double angle identity can be used to integrate other trigonometric functions, such as tanx, secx, and cscx. However, the specific steps and identities used may vary depending on the function being integrated.
Yes, when integrating using double angle, the key strategy is to use the double angle identities to simplify the expression being integrated. This often involves using trigonometric identities to rewrite the expression in terms of sine and cosine, and then applying the double angle identities to further simplify the expression.
Yes, the double angle identity can still be used to integrate expressions that include other variables. However, the specific steps and identities used may be more complex and may require additional algebraic manipulations.
Yes, there are some special cases when integrating using double angle, such as when the expression being integrated includes odd powers of sine and cosine. In these cases, the double angle identities may not be applicable, and other methods, such as integration by parts, may need to be used.