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eszkortwxp

Answered question

2022-04-24

My question is, how can I evaluate the following integral?

\int \frac{1}{\sqrt[4]{1 + x^{4}}}

Cloplerotly1gu

Beginner2022-04-25Added 11 answers

Let

x^{2} = \tan θ

. So

2 x d x = \sec^{2} θ d θ

. Carry out the substitution as usual, and for the sake of familiarity express the trigonometric functions in terms of

\sin

and

\cos

. I think we get

\int \frac{d θ}{2 \cos θ \sqrt{\sin θ}}

Multiply top and bottom by

\cos θ

. We get

\int \frac{\cos θ, d θ}{2 (1 - \sin^{2} θ) \sqrt{\sin θ}}

The natural substitution

y = \sin θ

gets us to

\int \frac{d y}{2 (1 - y^{2}) \sqrt{y}}

Now let

z = \sqrt{y}

. Then

d y = 2 z d z

, and we end up with

\int \frac{d z}{1 - z^{4}}

Now the game is about over, we use partial fractions as usual.

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