Question

integral of sin2x?
full steps

Answer 1

Please refer to the attachment.

Answer 2

HEY MATE YOUR ANSWER IS HERE

integral of sin2x

integral of sin2xTo integrate sin2x, also written as ∫sin2x dx, and sin 2x, we usually use a u substitution to build a new integration in terms of u.

LET U = 2x

$\frac{du}{dx} = 2$

$dx = \frac{1}{2} du$

We rearrange to get an expression for dx in terms of u

$sin2x \: dx = sin \: u \: \frac{1}{2} \: du$

As you can see, we now have a new integration in terms of u, which means the same thing. We get this by replacing 2x with u, and replacing dx with ½ du

$\sin \: u \: \frac{1}{2} \: du = sin \: u \: \frac{1}{2} \: du$

We move the ½ outside of the integral as it is simply a multiplier.

$sin \: u \: = - \: cos \: u$

We now have a simple integration with sinu that we can solve easily.

$\frac{1}{2} \: sin \: u \: du = - \frac{1}{2} cos \: u$

We remember the ½ outside the integral sign and reintroduce it here.

$= - \frac{1}{2} cos \: 2x + 2$

Hence, this is the final answer -½cos2x + C, where C is the integration constant.

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