What is the definite integral of $3 sin2x$ from $x = 0$ to $x =2$ ?

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Answer 1 · 2014-09-25T04:44:29

$int_0^2 3sin(2x)dx$

Factor out the constant

$3int_0^2 sin(2x)dx$

Make a substitution

Let $u=2x$

$du=2 dx$

$(du)/2=(2 dx)/2$

$(du)/2=dx$

$3int sin(u)*(du)/2$

$=3*1/2int sin(u)du$

$=3/2int sin(u)du$

Recalculate the boundaries

Upper Bound

$u=2x=2(2)=4$

Lower Bound

$u=2x=2(0)=0$

$3/2int_0^4 sin(u)du$

$=3/2[-cos(u)]_0^4$

$=3/2[-cos(4)-(-cos(0))]$

$=3/2[-cos(4)-(-1)]$

$=3/2[-cos(4)+1]$

$=[(-3cos(4)+3)/2]$

$=2.480465431$ sq units

What is the definite integral of 3 sin2x3 sin2x from x = 0x = 0 to x =2x =2?