Evaluate the integral: integral x tan x dx.

Question:

Evaluate the integral:

{eq}\displaystyle \int x \tan x\ dx {/eq}.

Indefinite Integrals:

Indefinite integrals or anti derivatives of a function {eq}fn {/eq} is a function {eq}F{/eq}; whose derivative is equal to {eq}fn {/eq}. We can display an indefinite integral like the given problem statement with an integral symbol, a function and then a differential at the end.

Answer and Explanation:

Given: $$\displaystyle \int x \tan x\ dx $$

The given expression can be solved using a non-elementary integral:

$$\int x \tan x\ dx =i\dfrac{1}{2}Li_{2}\left(-e^{2ix}\right)+i\dfrac{1}{2}x^{2}-x\ln\left(1+e^{2ix}\right) $$

Add the constant of integration.

$$\displaystyle =i\dfrac{1}{2}Li_{2}\left(-e^{2ix}\right)+i\dfrac{1}{2}x^{2}-x\ln\left(1+e^{2ix} \right) + C $$

Finally, we get integral as:

$$= \boxed{i\dfrac{1}{2}Li_{2} \left(-e^{2ix}\right)+i\dfrac{1}{2}x^{2}-x\ln\left(1+e^{2ix} \right) + C} $$


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Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

Chapter 12 / Lesson 11
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