How do you integrate #int x^2 tan2 x dx # using integration by parts?

Answer 1

You'll need more than just integration by parts. The integral involves the polylogarithm function evaluated at imaginary arguments..

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Answer 2

Christopher Agresta

To integrate ( \int x^2 \tan(2x) , dx ) using integration by parts, follow these steps:

Choose ( u = x^2 ) and ( dv = \tan(2x) , dx ).
Compute ( du = 2x , dx ) and ( v = \int \tan(2x) , dx = -\frac{1}{2} \ln|\cos(2x)| ).
Apply the integration by parts formula: ( \int u , dv = uv - \int v , du ).
Substitute the values of ( u ), ( dv ), ( du ), and ( v ) into the integration by parts formula.
Simplify the expression and integrate.

Applying these steps, you'll obtain the result:

[ \int x^2 \tan(2x) , dx = -\frac{1}{2}x^2 \ln|\cos(2x)| + \int x \ln|\cos(2x)| , dx ]

The integral ( \int x \ln|\cos(2x)| , dx ) may require further techniques such as integration by parts or substitution for evaluation, depending on the specific requirements of the problem.

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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.