Expert Answer to "Integrating 1sin⁡(3x) when sovling an ODE"

Jessie Jenkins

Answered question

2022-01-26

Integrating

\frac{1}{\sin (3 x)}

when sovling an ODE

Answer & Explanation

Jaiden Conrad

Beginner2022-01-27Added 14 answers

I = \int \frac{1}{\sin(3x)}dx
Let u=3x to yield:

I = \int \frac{1}{\sin (3 x)} d x = \int \frac{1}{\sin (u)} \cdot \frac{1}{3} d u = \frac{1}{3} \int \cos e c (u) d u

There are a variety of ways to approach this integral. One method is covered here. You will observe that method requires knowledge of a trigonometric derivative identity. Here I will employ a method that can be used to solve integrals of rational expressions of trigonometric functions. This method is known as the Half Tangent (aka Weierstrass) Substitution. Thus we let

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

to yield:

I = \frac{1}{3} \int \cos e c (u) d u = \frac{1}{3} \int \frac{1}{\sin (u)} d u

= \frac{1}{3} \int \frac{1}{\frac{2 t}{1 + t^{2}}} \frac{2}{1 + t^{2}} d t = \frac{1}{3} \int \frac{1}{t} : d t = \frac{1}{3} \ln | t | + C

= \frac{1}{3} \ln | : \tan (\frac{u}{2}) | + C = \frac{1}{3} \ln | \tan (\frac{3 x}{2}) | + C

Where C is the constant of integration.

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