\( \int \! \sin^n x \ dx \)       where \( n \in \mathbb{N} \) and \( n \ge 3 \)

\( \int \! \sin^n x \ dx = \int \! \left( \sin^{n-1} x \right) \sin x \ dx \)

\( u = \sin^{n-1} x \)

\( du = \left(n-1 \right) \sin^{n-2} x \cos x \ dx \)

\( dv = \sin x \ dx \)

\( v = – \cos x \)

\( \int \! \left( \sin^{n-1} x \right) \sin x \ dx = -\cos x \sin^{n-1} x + \left( n-1 \right) \int \! \sin^{n-2} x \cos^2 x \ dx \)

\( \int \! \left( \sin^{n-1} x \right) \sin x \ dx = -\cos x \sin^{n-1} x + \left( n-1 \right) \int \! \sin^{n-2} x \left( 1 – \sin^2 x \right) \ dx \)

\( \int \! \left( \sin^{n-1} x \right) \sin x \ dx = -\cos x \sin^{n-1} x + \left( n-1 \right) \int \! \sin^{n-2} x – \sin^{\left(n-2\right)+2} x \ dx \)

\( \int \! \left( \sin^{n-1} x \right) \sin x \ dx = -\cos x \sin^{n-1} x + \left( n-1 \right) \int \! \sin^{n-2} x – \sin^n x \ dx \)

\( \int \! \left( \sin^{n-1} x \right) \sin x \ dx = -\cos x \sin^{n-1} x + \left( n-1 \right) \int \! \sin^{n-2} x \ dx – \left(n-1 \right) \int \! \sin^n x \ dx \)

\( \int \! \left( \sin^{n-1} x \right) \sin x \ dx + \left(n-1 \right) \int \! \sin^n x \ dx= -\cos x \sin^{n-1} x + \left( n-1 \right) \int \! \sin^{n-2} x \ dx \)

\( n \int \! \left( \sin^{n-1} x \right) \sin x \ dx = -\cos x \sin^{n-1} x + \left( n-1 \right) \int \! \sin^{n-2} x \ dx \)

\( \int \! \left( \sin^{n-1} x \right) \sin x \ dx = \frac{1}n \left( -\cos x \sin^{n-1} x + \left( n-1 \right) \int \! \sin^{n-2} x \ dx \right) \)

 

\( \int \! \cos^n x \ dx \)       where \( n \in \mathbb{N} \) and \( n \ge 3 \)

\( \int \! \cos^n x \ dx = \int \! \left( \cos^{n-1} x \right) \cos x \ dx \)

\( u = \cos^{n-1} x \)

\( du = – \left(n-1 \right) \cos^{n-2} x \sin x \ dx \)

\( dv = \cos x \ dx \)

\( v = \sin x \)

\( \int \! \left( \cos^{n-1} x \right) \cos x \ dx = \sin x \cos^{n-1} x + \left( n-1 \right) \int \! \cos^{n-2} x \sin^2 x \ dx \)

\( \int \! \left( \cos^{n-1} x \right) \cos x \ dx = \sin x \cos^{n-1} x + \left( n-1 \right) \int \! \cos^{n-2} x \left( 1 – \cos^2 x \right) \ dx \)

\( \int \! \left( \cos^{n-1} x \right) \cos x \ dx = \sin x \cos^{n-1} x + \left( n-1 \right) \int \! \cos^{n-2} x – \cos^{\left(n-2\right)+2} x \ dx \)

\( \int \! \left( \cos^{n-1} x \right) \cos x \ dx = \sin x \cos^{n-1} x + \left( n-1 \right) \int \! \cos^{n-2} x – \cos^n x \ dx \)

\( \int \! \left( \cos^{n-1} x \right) \cos x \ dx = \sin x \cos^{n-1} x + \left( n-1 \right) \int \! \cos^{n-2} x \ dx – \left(n-1 \right) \int \! \cos^n x \ dx \)

\(n \int \! \left( \cos^{n-1} x \right) \cos x \ dx = \sin x \cos^{n-1} x + \left( n-1 \right) \int \! \cos^{n-2} x \ dx \)

\( \int \! \left( \cos^{n-1} x \right) \cos x \ dx = \frac{1}n \left( \sin x \cos^{n-1} x + \left( n-1 \right) \int \! \cos^{n-2} x \ dx \right) \)

 

\( \int \! \tan^n x \ dx \)       where \( n \in \mathbb{N} \) and \( n \ge 2 \)

\( \int \! \tan^n x \ dx = \int \! \tan^{n-2}x \tan^2 x \ dx\)

\( \int \! \tan^n x \ dx = \int \! \tan^{n-2}x \left( \sec^2 x – 1 \right) dx\)

\( \int \! \tan^n x \ dx = \int \! \tan^{n-2}x \sec^2 x \ dx – \int \! \tan^{n-2}x \ dx \)

\( \int \! \tan^{n-2}x \sec^2 x \ dx\)

\( u = \tan x \)

\( du = \sec^2 x \ dx\)

\( \int \! u^{n-2} \ du = \frac{u^{n-1}}{n-1} = \frac{ \tan^{n-1}x}{n-1} \)

\( \int \! \tan^n x \ dx = \frac{ \tan^{n-1}x}{n-1} – \int \! \tan^{n-2}x \ dx \)

 

\( \int \! \sec^n x \ dx \)       where \( n \in \mathbb{N} \) and \( n \ge 3 \)

\( \int \! \sec^n x \ dx = \int \! \sec^2 x \left( \sec^{n-2} x \right) \ dx \)

\( u = \sec^{n-2} x \)

\( du = \left(n-2 \right) \sec^{n-3} x \sec x \tan x \ dx \)

\( dv = \sec^2 x \ dx \)

\( v = \tan x \)

\( \int \! \sec^2 x \left( \sec^{n-2} x \right) \ dx = \tan x \sec^{n-2} x – \left( n-2 \right) \int \! \tan^2 x \sec^{n-2} x \ dx \)

\( \int \! \sec^2 x \left( \sec^{n-2} x \right) \ dx = \tan x \sec^{n-2} x – \left( n-2 \right) \int \! \left( \sec^2 x – 1 \right) \sec^{n-2} x \ dx \)

\( \int \! \sec^2 x \left( \sec^{n-2} x \right) \ dx = \tan x \sec^{n-2} x – \left( n-2 \right) \int \! \sec^n x – \sec^{n-2} x \ dx \)

\( \int \! \sec^2 x \left( \sec^{n-2} x \right) \ dx = \tan x \sec^{n-2} x – \left( n-2 \right) \int \! \sec^n x \ dx + \left(n-2 \right) \int \! \sec^{n-2} x \ dx \)

\( \left(n-1 \right) \int \! \sec^2 x \left( \sec^{n-2} x \right) \ dx = \tan x \sec^{n-2} x + \left( n-2 \right) \int \! \sec^{n-2} x \ dx \)

\( \int \! \sec^2 x \left( \sec^{n-2} x \right) \ dx = \frac{1}{n-1} \left( \tan x \sec^{n-2} x + \left( n-2 \right) \int \! \sec^{n-2} x \ dx \right) \)

 

\( \int \! \csc^n x \ dx \)       where \( n \in \mathbb{N} \) and \( n \ge 3 \)

\( \int \! \csc^n x \ dx = \int \! \csc^2 x \left( \csc^{n-2} x \right) \ dx \)

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

\( du = \left(n-2 \right) \csc^{n-3} x \left(-\csc x \cot x \right) \ dx \)

\( dv = \csc^2 x \ dx \)

\( v = -\cot x \)

\( \int \! \csc^2 x \left( \csc^{n-2} x \right) \ dx = -\cot x \csc^{n-2} x – \left( n-2 \right) \int \! \cot^2 x \csc^{n-2} x \ dx \)

\( \int \! \csc^2 x \left( \csc^{n-2} x \right) \ dx = -\cot x \csc^{n-2} x – \left( n-2 \right) \int \! \left( \csc^2 x – 1 \right) \sec^{n-2} x \ dx \)

\( \int \! \csc^2 x \left( \csc^{n-2} x \right) \ dx = -\cot x \csc^{n-2} x – \left( n-2 \right) \int \! \csc^n x – \csc^{n-2} x \ dx \)

\( \int \! \csc^2 x \left( \csc^{n-2} x \right) \ dx = -\cot x \csc^{n-2} x – \left( n-2 \right) \int \! \csc^n x \ dx + \left(n-2 \right) \int \! \csc^{n-2} x \ dx \)

\( \left(n-1 \right) \int \! \csc^2 x \left( \csc^{n-2} x \right) \ dx = -\cot x \csc^{n-2} x + \left( n-2 \right) \int \! \csc^{n-2} x \ dx \)

\( \int \! \csc^2 x \left( \csc^{n-2} x \right) \ dx = \frac{1}{n-1} \left( -\cot x \csc^{n-2} x + \left( n-2 \right) \int \! \csc^{n-2} x \ dx \right) \)