you can once you find that the derivative of your substitution appears in your original problemtunabeast said:fikus what do you mean, i can do a substitution straight away?
I believe that is what everyone as been trying to tell you!tunabeast said:fikus what do you mean, i can do a substitution straight away?
The antiderivative of sin^3(x)cos^4(x) is (1/12)sin^4(x) - (1/8)sin^2(x) + (1/48)cos^6(x) + C.
The antiderivative can be derived by using the substitution method and the power reduction formula for sin^2(x).
Example: Find the antiderivative of sin^3(x)cos^4(x).
Solution: Let u = sin(x), du = cos(x)dx. Then, we have sin^3(x)cos^4(x)dx = u^3(1-u^2)^2du. Using the power reduction formula for sin^2(x), we get u^3(1-(1/2)sin^2(x))^2du. Expanding and simplifying, we get (1/12)sin^4(x) - (1/8)sin^2(x) + (1/48)cos^6(x)+ C, where C is the constant of integration.
Yes, the antiderivative of sin^3(x)cos^4(x) is unique up to a constant of integration.
No, the antiderivative of sin^3(x)cos^4(x) cannot be expressed in terms of other trigonometric functions. It can only be expressed as a combination of powers of sin(x) and cos(x).