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What is the integral of #int tan(2x) dx#?

Answer 1

Evelyn Agard

#inttan(2x)dx =-1/2ln|cos(2x)|+C#

We will proceed using substitution, along with the fact that #int1/xdx = ln|x|+C#

Let #u = cos(2x) => du = -2sin(2x)dx#

Next, we have

#inttan(2x)dx = intsin(2x)/cos(2x)dx#

#=-1/2int1/cos(2x)(-2sin(2x))dx#

#=-1/2int1/udu#

#=-1/2ln|u|+C#

#=-1/2ln|cos(2x)|+C#

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Answer 2

Adam Adkins

The integral of ( \int \tan(2x) , dx ) is ( -\frac{1}{2} \ln|\cos(2x)| + C ), where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.