secant Archives - Mr. Kennedy: Another Maths Enthusiast

$ \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) $

In order to find the derivatives of the trigonometric functions the definition of the derivative must be used. Each of the following cases uses the form

$$ f’\left(x \right) = \lim_{h \rightarrow 0} \frac{f \left( x + h \right) – f \left( x \right)}{h} $$

Let $ f \left(x \right) = \sin \left( x \right) $

$$ \lim_{h \rightarrow 0} \frac{ \sin \left( x + h \right) – \sin \left( x \right)}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin \left( x \right) \cos \left( h \right) + \sin \left( h \right) \cos \left( x \right) – \sin \left( x \right)}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin \left( x \right) \left( \cos h – 1 \right) + \cos \left( x \right) \sin \left( h \right)}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin \left( x \right) \left( \cos h – 1 \right)}{h} + \frac{\cos \left( x \right) \sin \left( h \right)}{h} = \cos \left( x \right) $$

Let $ f \left(x \right) = \cos x $

$$ \lim_{h \rightarrow 0} \frac{ \cos \left( x + h \right) – \cos x}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \cos x \cos h – \sin h \sin x – \cos x}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \cos x \left( \cos h – 1 \right) – \sin h \sin x}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \cos x \left( \cos h – 1 \right)}{h} – \frac{\sin x \sin h}{h} = – \sin x $$

Let $ f \left(x \right) = \tan x $

$$\lim_{h \rightarrow 0} \frac{ \tan \left( x + h \right) – \tan x}{h}$$

$$\lim_{h \rightarrow 0} \frac{ \frac{\sin \left(x + h \right)}{\cos \left(x + h \right)} – \frac{ \sin x}{\cos x}}{h}$$

$$ \lim_{h \rightarrow 0} \frac{\sin \left(x + h \right) \cos x – \sin x \left( \cos \left( x + h \right) \right)}{h \cos \left(x + h \right) \cos x} $$

$$ \lim_{h \rightarrow 0} \frac{\left( \sin x \cos h + \sin h \cos x \right) \cos x – \sin x \left( \cos x \cos h – \sin x \sin h \right)}{h \cos \left(x + h \right) \cos x} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin h \cos^2 x}{h \cos \left(x + h \right) \cos x} + \frac{\sin^2 x \sin h}{h \cos \left(x + h \right) \cos x}= 1 + \tan^2 x = \sec^2 x $$

Let $ f \left(x \right) = \csc x $

$$ \lim_{h \rightarrow 0} \frac{ \csc \left( x + h \right) – \csc x}{h} $$

$$ \lim_{h \rightarrow 0} \frac{ \frac{1}{\sin \left(x + h \right)} – \frac{1}{\sin x}}{h} $$

$$ \lim_{h \rightarrow 0} \frac{\sin x – \sin \left( x + h \right)}{h \sin \left(x + h \right) \sin x} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin x – \sin x \cos h – \sin h \cos x}{h \sin \left(x + h \right) \sin x} $$

$$ \lim_{h \rightarrow 0} \frac{ \sin x \left( 1 – \cos h \right)}{h \sin \left(x + h \right) \sin x} – \frac{\sin h \cos x}{h \sin \left(x + h \right) \sin x}= – \cot x \csc x $$

Let $ f \left(x \right) = \sec x $

$$\lim_{h \rightarrow 0} \frac{ \sec \left( x + h \right) – \sec x}{h}$$

$$\lim_{h \rightarrow 0} \frac{ \frac{1}{\cos \left(x + h \right)} – \frac{1}{\cos x}}{h}$$

$$\lim_{h \rightarrow 0} \frac{\cos x – \cos \left( x + h \right)}{h \cos \left(x + h \right) \cos x}$$

$$\lim_{h \rightarrow 0} \frac{ \cos x – \cos x \cos h + \sin h \sin x}{h \cos \left(x + h \right) \cos x}$$

$$\lim_{h \rightarrow 0} \frac{ \cos x \left( 1 – \cos h \right)}{h \cos \left(x + h \right) \cos x} + \frac{\sin h \sin x}{h \cos \left(x + h \right) \cos x}= \tan x \sec x$$

Mr. Kennedy: Another Maths Enthusiast

I like to teach, learn and study mathematics & physics

Tag Archives: secant

Reduction Formulas

Derivatives of Some Trig Functions