"Evaluate the following integrals. ∫(csc2x+csc4x)dx" was answered by our community

BolkowN

Answered question

2021-10-10

Evaluate the following integrals.

\int (\csc^{2} x + \csc^{4} x) d x

Mayme

Skilled2021-10-11Added 103 answers

We know that,

\csc^{2} x = 1 + \cot^{2} x

We have,

I = \int (\csc^{2} x + \csc^{4} x) d x

On solving further, we get our result as

I = \int (\csc^{2} x + \csc^{4} x) d x

= \int (\csc^{2} x) (1 + \csc^{2} x) d x

= \int (\csc^{2} x) (1 + 1 + \cot^{2} x) d x

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

Let us consider,

\cot x = u

(- \csc^{2} x) d x = d u

(\csc^{2} x) d x = - d u

On substituting

(\csc^{2} x) d x = - d u

, integral becomes as

I = \int (2 + u^{2}) (- d u)

I = - \int 2 d u - \int u^{2} d u

I = - 2 u - \frac{u^{3}}{3} + C

On substituting back

\cot x = u

, we get our result as

I = - 2 \cot x - \frac{\cot^{3} x}{3} + C

Hence, value of integral

\int (\csc^{2} x + \csc^{4} x) d x

- 2 \cot x - \frac{\cot^{3} x}{3} + C

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