Question

ntegrate sin(2x) dx why is the answer -1/2 cos(2x)+2 why the 1/2

Answer 1

yeah your answer is right and the 1/2 is there so that when you differentiate your answer you can get back to sin(2x) and its negative because the derivative of cos(x)=-sin(x)

Answer 2

why 1/2?

Answer 3

is there like a formula?

Answer 4

are you familiar with the u-substitution? because i'm going to use it...if you don't know it just tell me and i can solve it by the basic method ;) personally i prefer not to because it confuses me lol anyway...

\[\int \sin (2x) dx\]
let u = 2x \(\rightarrow\) du = 2dx \(\rightarrow \frac{du}{2} = dx\)

\[\int \sin (2x)dx \implies \int \sin (u) \frac{du}{2}\]
\[\implies \frac12 \int \sin u du\]

integral of sin u is -cos u therefore

\[\frac 12 \int \sin u du = -\frac{1}{2} \cos u\] but dont forget to sub back

\[-\frac{1}{2} \cos u \implies -\frac 12 \cos (2x)\]

Answer 5

if you're not familiar with u-sub...i'll use the good old messy method

\[\int \sin (2x) dx\]
this is not a perfect integral you need a 2dx for it to be a perfect integral

\[\int [\sin (2x)] (2dx)\]
now it's a perfect integral..but you need to multiply by 1/2 because you multiplied by 2 (so that nothing will change in terms of value)

\[\frac{1}{2} \int \sin (2x) (2dx)\]

the integral of that is just \(-\cos (2x)\)

so...

\[\frac 12 \int \sin (2x) (2dx) \implies -\frac 12 \cos (2x)\]

u-sub is easier to understand isnt it. lol

Answer 6

do you get either of the methods i wrote @wilbert06 ?

Answer 7

@igbasallote I dont quite fully understand them maybe if you would do an examample I could see it

Answer 8

or u actually did

Answer 9

yes i do thank you

Answer 10

for future references...it's Lgbasallote not igbasallote ;) and glad to help ^_^

Answer 11

haha thank you!