Why are we forced to use the double angle formula to integrate $\int 2\sin x \cos x\ \mathrm{d}x$

Question

I tried it using the double angle identity $$\sin{2x}=2\sin x\cos x$$

The answer that I got is $$\frac{-\cos 2x}{4} +c$$ However I've also tried it using $u$-substitution.

I let $u=\sin x$. Thus obtaining $\cos x$ when differentiating. And cutting the $\cos x$ in $2\sin x\cos x$ out with the $\cos x$ in the denominator below $du$.

However the answer that I am then getting is : $0.25 - 0.25\cos 2x + c$. So as you can see there is the extra term $0.25$ there. Is the second answer deemed to be wrong? If so why? My book tells me to use the double angle formula but does not explain why.

Both approaches are correct. The $.25$ can be absorbed into the "$+C$". In other words, the two antiderivatives you found differ only by a constant, and that is to be expected. — littleO, Apr 01 '21 at 04:54
Both of your anti-derivatives are correct. This is explained by the fact that they only differ by a constant. As your analysis indicates, you can use a substitution, rather than the double angle formula. However, the book's thinking is probably that the double angle formula is easier, because just as you immediately know how to integrate $\int \sin x$, you can also easily integrate $\int \sin(kx)$. — user2661923, Apr 01 '21 at 04:55
I see. But what if I had limits of integration? Wouldn't I get two different answers? — butterfly, Apr 01 '21 at 04:59
Nope! Try it and see: Note that if $F'(x) = f(x)$, and $G(x) = F(x) + C$, then $\int_a^b f(x)dx = F(b) - F(a) = G(b) - G(a)$. — user2661923, Apr 01 '21 at 05:01
Does this answer your question? Simple u-subsitution - Paradoxical Result — Toby Mak, Apr 01 '21 at 05:16
Does this answer your question? Trig integral $\int ( \cos{x} + \sin{x}\cos{x}) , dx $ — Hans Lundmark, Apr 01 '21 at 14:22

score 4 · Accepted Answer · edited Apr 01 '21 at 04:57

4

Both answers are correct. You can use double angle identity, as well as u sub for either $\sin x$ or $\cos x$.

The key lies in the +c. All the 3 integrals are a family of functions just separated by a different "+c". In practice, double angle identity is often used as it's more intuitive and simpler in some sense. But the other methods are perfectly acceptable, and not "wrong."

edited Apr 01 '21 at 04:57

Ishraaq Parvez

1,731

answered Apr 01 '21 at 04:54

devam_04

748

But you say that the constant of integration changes. Suppose i had limits of integration (let us say 1 to 2 ) for this question. My answer would be different, right? – butterfly Apr 01 '21 at 04:57
It won't as the constant would get subtracted while plugging in the upper and lower bounds. – devam_04 Apr 01 '21 at 05:00
Ohh.. OK. Thanks for the help. I am approving your answer. – butterfly Apr 01 '21 at 05:06

score 1 · Answer 2 · answered Apr 01 '21 at 04:55

1

Actually your method is not wrong.

All you need to do is just substitute \begin{equation*} c_{2} =0.25+c \end{equation*} and then you get the "correct" answer.

answered Apr 01 '21 at 04:55

Ishraaq Parvez

1,731

score 1 · Answer 3 · answered Apr 01 '21 at 05:11

1

$$\frac{1}{2} \frac{d}{dx} \sin^2 x = \sin x \cos x$$ so $$\int \sin x \cos x \,dx = \frac{1}{2} \sin^2 x + \text{constant}$$

answered Apr 01 '21 at 05:11

mjw

8,647
1
8
23

score 1 · Answer 4 · answered Apr 01 '21 at 05:39

Try graphing the two answers. You will see that they have equal slopes at each $x$ value, so they are both anti-derivatives of the same thing. Moving either vertically up or down does not change this (and can make them coincide). That is the $+C$, an arbitrary constant of integration.

Why are we forced to use the double angle formula to integrate $\int 2\sin x \cos x\ \mathrm{d}x$

4 Answers4