"How to integrate csc⁡(x)" was answered by students like you

Karen Simpson

Answered question

2021-12-13

How to integrate

\csc (x)

nghodlokl

Beginner2021-12-14Added 33 answers

\int \csc (x) d x = \int \csc (x) \frac{\csc (x) + \cot (x)}{\csc (x) + \cot (x)} d x = \int \frac{\csc^{2} (x) + \csc (x) \cot (x)}{\csc (x) + \cot (x)} d x

Let

u = \csc (x) + \cot (x) \Rightarrow \frac{d u}{d x} = - \csc (x) \cot (x) - \csc^{2} (x) = - (\csc (x) \cot (x) + \csc^{2} (x))

Thus,

\int \csc (x) d x = \int (- \frac{1}{u}) d u = - \int \frac{1}{u} d u = - \ln | u | + C = - \ln | \csc (x) + \cot (x) | + C

autormtak0w

Beginner2021-12-15Added 31 answers

\int \csc (x) d x

Apply u-substitution:

u = \tan (\frac{x}{2})

\int \csc (x) d x = \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = \ln | u |

Substitute back

= \ln | \tan (\frac{x}{2}) |

= \ln | \tan (\frac{x}{2}) | + C

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