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Dick said:Stop at the third line. Now move the integral of cos(x)*e^x on the right to the left. Then you are basically done. The third line gives you an equation where the only integral is cos(x)*e^x. Solve for it.
theBEAST said:Oh I see, I got
∫(e^xcosx dx)=(e^xsinx+e^xcosx)/2
But how did I get :
∫(e^xcosx dx)=e^xsinx+e^xcosx-∫(e^xcosx dx) (in the third step)
and
∫(e^xcosx dx)=∫(e^xcosx dx) (in the last step)
The two are not the same thing?
The integral of e^x(cosx) is equal to e^x(sin(x) + cos(x)) + C, where C is the constant of integration.
To solve the integral of e^x(cosx), you can use the technique of integration by parts, where u = cosx and dv = e^x dx. After integration by parts, you will get e^x(sin(x) + cos(x)) + C.
Yes, the integral of e^x(cosx) can also be solved using substitution. You can let u = sin(x) + cos(x), then du = (cos(x) - sin(x))dx. After substitution, the integral will become e^x(u)du, which can be solved by integrating by parts.
The difference between the integral of e^x(cosx) and e^(x^2) is that e^x(cosx) has a trigonometric function, while e^(x^2) does not. The integral of e^(x^2) can be solved using the substitution method, while the integral of e^x(cosx) requires integration by parts.
Yes, there is a general formula for the integral of e^x(cosx), which is e^x(sin(x) + cos(x)) + C. This formula can be applied to any integral of the form e^x(f(x)), where f(x) is a trigonometric function.