math bug.. integral of (sin 2x) and integral of (2 sin x cos x) are different...

[DuctTapePro]

math bug: integral of (sin 2x) and integral of (2 sin x cos x) are different.

why they give different results? they're trigonometric identities right?

 

[Deleted member 4993]

why they give different results? they're trigonometric identities right?

I am assuming that the two different answers that you are getting are

-1/2 * cos(2x) and

-cos²(x)

As you know those differ by a constant [cos²(x) - 1/2 * cos(2x) = 1/2]. Those constants of integrations, though very important, do not affect the final result further manipulations.

 

[Dr.Peterson]

why they give different results? they're trigonometric identities right?

As Khan said (sort of), your two answers are presumably -cos(2x)/2 and sin^2(x). But really they should be written as -cos(2x)/2 + C₁ and sin^2(x) + C₂. Those arbitrary constants are essential.

When you find that your answer to an integral looks different from someone else's (especially the book's), you can try to rewrite yours in terms of theirs. Starting with -cos(2x)/2 + C₁, we can apply the double-angle identity and get -(1 - 2 sin²(x))/2 + C₁ = sin²(x) - 1/2 + C₁. Since C1 can be any constant, we can "absorb" the -1/2 into it to make a new arbitrary constant, which is exactly your other answer: sin^2(x) + C₂.

Another thing you can do when your answer is different from the book's is to differentiate each of them. If they both yield the integrand, then both are right. (Sometimes the book is wrong.)