Find out the answer to "How do I integrate the following:∫tan⁡ax1+tan⁡axdx"

Elijah Schwartz

Answered question

2022-04-13

How do I integrate the following:

\int \frac{\sqrt{\tan a x}}{1 + \sqrt{\tan a x}} d x

folytonr4qx

Beginner2022-04-14Added 14 answers

Hint:
Substituting u=ax, gives us:

I = \int \frac{\sqrt{\tan a x}}{1 + \sqrt{\tan a x}} d x

I = \frac{1}{a} \int \frac{\sqrt{\tan u}}{1 + \sqrt{\tan u}} d u

a I = \int d u - \int \frac{1}{\sqrt{\tan u} + 1} d u

Now the second integral cam be written as:

I_{1} = \int \frac{1}{1 + \sqrt{\tan u}} d u

= 2 \int \frac{v - 1}{v ({(v - 1)}^{4} + 1)} d v

after substituting

v = 1 + \sqrt{\tan u}

The next step is obviously a PFD

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