Inverse Trigonometric function

1. DOMAIN, RANGE & GRAPH OFINVERSE TRIGONOMETRIC

FUNCTIONS :

(a)

Graph of arc sin(x) or graph of sin^(-1)x

$\mathrm{f}^{-1}:[-1,1] \rightarrow[-\pi / 2, \pi / 2]$ ,

$\mathrm{f}^{-1}(\mathrm{x})=\sin ^{-1}(\mathrm{x})$

(b)

Graph of arc cos(x) or graph of cos^(-1)x

$\mathrm{f}^{-1}:[-1,1] \rightarrow[0, \pi]$ ,

$\mathrm{f}^{-1}(\mathrm{x})=\cos ^{-1} \mathrm{x}$

(c)

$\mathrm{f}^{-1}: \mathrm{R} \rightarrow(-\pi / 2, \pi / 2)$ ,

$\mathrm{f}^{-1}(\mathrm{x})=\tan ^{-1} \mathrm{x}$

(d)

$\mathrm{f}^{-1}: \mathrm{R} \rightarrow(0, \pi)$

$\mathrm{f}^{-1}(\mathrm{x})=\cot ^{-1} \mathrm{x}$

(e)

$\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)$ $\rightarrow[0, \pi / 2) \cup(\pi / 2, \pi]$ ,

$\mathrm{f}^{-1}(\mathrm{x})=\sec ^{-1} \mathrm{x}$

(f)

$\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)$ $\rightarrow[-\pi / 2,0) \cup(0, \pi / 2]$

$\mathrm{f}^{-1}(\mathrm{x})=\operatorname{cosec}^{-1} \mathrm{x}$

All Chapter Notes, Concept and Important Formula

2. PROPERTIES OF INVERSE CIRCULAR FUNCTIONS:

Property-1 :

(i) $y=\sin \left(\sin ^{-1} x\right)=x, \quad x \in[-1,1], y \in[-1,1], y$ is aperiodic

(ii) $y=\cos \left(\cos ^{-1} x\right)$ $=x \quad, x \in[-1,1],$ $y\,\, \in \,[-1,1], y$ is aperiodic

(iii) $y=\tan \left(\tan ^{-1} x\right)=x, x \in R, y \in R, y$ is aperiodic

(iv) $y=\cot \left(\cot ^{-1} x\right)=x, x \in R, y \in R, y$ is aperiodic

(v) $y=\operatorname{cosec}\left(\operatorname{cosec}^{-1} x\right)=x,|x| \geq 1,|y| \geq 1, y$ is aperiodic

(vi) $y=\sec \left(\sec ^{-1} x\right)=x,|x| \geq 1 ;|y| \geq 1, y$ is aperiodic

Property 2 :

(i) $y=\sin ^{-1}(\sin x), x \in R, y \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] .$ Periodic with period $2 \pi$ .

$\sin ^{-1}(\sin x)=\left\{\begin{array}{cc}-\pi-x & ,-\frac{3 \pi}{2} \leq x \leq-\frac{\pi}{2} \\ x & , \quad-\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \\ \pi-x & , \quad \frac{\pi}{2} \leq x \leq \frac{3 \pi}{2} \\ x-2 \pi & , \quad \frac{3 \pi}{2} \leq x \leq \frac{5 \pi}{2} \\ 3 \pi-x & , \quad \frac{5 \pi}{2} \leq x \leq \frac{7 \pi}{2} \\ x-4 \pi & , \quad \frac{7 \pi}{2} \leq x \leq \frac{9 \pi}{2}\end{array}\right.$

(ii) $y=\cos ^{-1}(\cos x), x \in R, y \in[0, \pi]$ , periodic with period $2 \pi$

$\cos ^{-1}(\cos x)=\left\{\begin{array}{ccc}-x & , & -\pi \leq x \leq 0 \\ x & , & 0 \leq x \leq \pi \\ 2 \pi-x & , & \pi \leq x \leq 2 \pi \\ x-2 \pi & , & 2 \pi \leq x \leq 3 \pi \\ 4 \pi-x & , & 3 \pi \leq x \leq 4 \pi\end{array}\right.$

(iii) $y=\tan ^{-1}(\tan x)$ $\mathrm{x} \in \mathrm{R}-\left\{(2 \mathrm{n}-1) \frac{\pi}{2}, \mathrm{n} \in \mathrm{I}\right\};$ $y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ , periodic with period $\pi$

$\tan ^{-1}(\tan x)=\left\{\begin{array}{cc}x+\pi & ,-\frac{3 \pi}{2}<x<-\frac{\pi}{2} \\ x & , \quad-\frac{\pi}{2}<x<\frac{\pi}{2} \\ x-\pi & , \quad \frac{\pi}{2}<x<\frac{3 \pi}{2} \\ x-2 \pi & , \quad \frac{3 \pi}{2}<x<\frac{5 \pi}{2} \\ x-3 \pi & , \quad \frac{5 \pi}{2}<x<\frac{7 \pi}{2}\end{array}\right.$

(iv) $\mathrm{y}=\cot ^{-1}(\cot \mathrm{x}), \mathrm{x} \in \mathrm{R}-\{\mathrm{n} \pi\}, \mathrm{y} \in(0, \pi)$ , periodic with period $\pi$

(v) $y=\operatorname{cosec}^{-1}(\operatorname{cosec} x),$ $x \in R-\{n \pi, n \in I\}$ $y \in\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]$ , $y$ is periodic with period $2 \pi$ .

(vi) $\mathrm{y}=\sec ^{-1}(\sec \mathrm{x}), \mathrm{y}$ is periodic with period $2 \pi$

$\mathrm{x} \in \mathrm{R}-\left\{(2 \mathrm{n}-1) \frac{\pi}{2} \mathrm{n} \in \mathrm{I}\right\}, \quad \mathrm{y} \in\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]$

Property 3 :

(i) $\operatorname{cosec}^{-1} \mathrm{x}=\sin ^{-1} \frac{1}{\mathrm{x}} ; \quad \mathrm{x} \leq-1$ or $\mathrm{x} \geq 1$

(ii) $\sec ^{-1} \mathrm{x}=\cos ^{-1} \frac{1}{\mathrm{x}} ; \quad \mathrm{x} \leq-1$ or $\mathrm{x} \geq 1$

(iii) $\cot ^{-1} \mathrm{x}=\tan ^{-1} \frac{1}{\mathrm{x}} ; \mathrm{x}>0$

$\qquad \qquad \quad=\pi+\tan ^{-1} \frac{1}{\mathrm{x}} ; \mathrm{x}<0$

Property 4:

(i) $\sin ^{-1}(-\mathrm{x})=-\sin ^{-1} \mathrm{x}, \quad-1 \leq \mathrm{x} \leq 1$

(ii) $\tan ^{-1}(-\mathrm{x})=-\tan ^{-1} \mathrm{x}, \mathrm{x} \in \mathrm{R}$

(iii) $\cos ^{-1}(-\mathrm{x})=\pi-\cos ^{-1} \mathrm{x},-1 \leq \mathrm{x} \leq 1$

(iv) $\cot ^{-1}(-\mathrm{x})=\pi-\cot ^{-1} \mathrm{x}, \mathrm{x} \in \mathrm{R}$

(v) $\sec ^{-1}(-x)=\pi-\sec ^{-1} x, x \leq-1$ or $x \geq 1$

(vi) $\operatorname{cosec}^{-1}(-\mathrm{x})=-\operatorname{cosec}^{-1} \mathrm{x}, \mathrm{x} \leq-1$ or $\mathrm{x} \geq 1$

Property 5 :

(i) $\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2},-1 \leq x \leq 1$

(ii) $\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, \quad x \in R$

(iii) $\operatorname{cosec}^{-1} x+\sec ^{-1} x=\frac{\pi}{2},|x| \geq 1$

Property 6:

(i) $\tan ^{-1} x+\tan ^{-1} y$ $=\left\{\begin{array}{l}\tan ^{-1} \frac{x+y}{1-x y}, \text { where } x>0, y>0 \, \&\, x y<1 \\ \pi+\tan ^{-1} \frac{x+y}{1-x y}, \text { where } x>0, y>0\, \&\, x y>1 \\ \frac{\pi}{2}, \text { where } x>0, y>0 \, \&\, x y=1\end{array}\right.$

(ii) $\tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}$ , where $x>0,$ $y>0$

(iii)

$\sin ^{-1} x+\sin ^{-1} y$

$=\sin ^{-1}\left[x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right]$
where

$x>0, y>0$ &

$\left(x^{2}+y^{2}\right)<1$

Note that : $\mathrm{x}^{2}+\mathrm{y}^{2}<1 \Rightarrow 0<\sin ^{-1} \mathrm{x}+\sin ^{-1} \mathrm{y}<\frac{\pi}{2}$

(iv)

$\sin ^{-1} x+\sin ^{-1} y$

$=\pi-\sin ^{-1}\left[x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right]$ ,
where

$x>0, y>0$ &

$x^{2}+y^{2}>1$

Note that : $\mathrm{x}^{2}+y^{2}>1 \Rightarrow \frac{\pi}{2}<\sin ^{-1} \mathrm{x}+\sin ^{-1} \mathrm{y}<\pi$

(v)

$\sin ^{-1} x-\sin ^{-1} y$

$=\sin ^{-1}\left[x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right]$ where

$x>0, y>0$

(vi) $\cos ^{-1} x+\cos ^{-1} y$ $=\cos ^{-1}\left[x y-\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right]$ , where $x>0, y>0$

(vii) $\cos ^{-1} x-\cos ^{-1} y=$ $\left\{\begin{array}{ll}\cos ^{-1}\left(x y+\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right); & x<y, \, x, y>0 \\ -\cos ^{-1}\left(x y+\sqrt{1-x^{2}} \sqrt{1-y^{2}}\right) ; & x>y, \, x, y>0\end{array}\right.$

(viii)

$\tan ^{-1} \mathrm{x}+\tan ^{-1} \mathrm{y}+\tan ^{-1} \mathrm{z}$

$=\tan ^{-1}\left[\dfrac{\mathrm{x}+\mathrm{y}+\mathrm{z}-\mathrm{xyz}}{1-\mathrm{x} y-\mathrm{yz}-\mathrm{zx}}\right]$
if

$x>0, y>0, z>0$ &

$x y+y z+z x<1$

Note : In the above results $\mathrm{x}$ & $\mathrm{y}$ are taken positive. In case if these are given as negative, we first apply $\mathrm{P}-4$ and then use above results.

3. SIMPLIFIED INVERSE TRIGONOMETRIC FUNCTIONS :

(a)

$y=f(x)=\sin ^{-1}\left(\dfrac{2 x}{1+x^{2}}\right)$

$=\left[\begin{array}{ccc}2 \tan ^{-1} x & \text { if } & \mid x \leq 1 \\ \pi-2 \tan ^{-1} x & \text { if } & x>1 \\ -\left(\pi+2 \tan ^{-1} x\right) & \text { if } & x<-1\end{array}\right.$

(b)

$y=f(x)=\cos ^{-1}\left(\dfrac{1-x^{2}}{1+x^{2}}\right)$

$=\left[\begin{array}{ccc}2 \tan ^{-1} x & \text { if } & x \geq 0 \\ -2 \tan ^{-1} x & \text { if } & x<0\end{array}\right.$

(c)

$y=f(x)=\tan ^{-1} \dfrac{2 x}{1-x^{2}}$

$=\left[\begin{array}{llc}2 \tan ^{-1} x & \text { if } & |x|<1 \\ \pi+2 \tan ^{-1} x & \text { if } & x<-1 \\ -\left(\pi-2 \tan ^{-1} x\right) & \text { if } & x>1\end{array}\right.$

(d)

$y=f(x)=\sin ^{-1}\left(3 x-4 x^{3}\right)$

$=\left[\begin{array}{ccc}-\left(\pi+3 \sin ^{-1} x\right) & \text { if } & -1 \leq x \leq-\frac{1}{2} \\ 3 \sin ^{-1} x & \text { if } & -\frac{1}{2} \leq x \leq \frac{1}{2} \\ \pi-3 \sin ^{-1} x & \text { if } & \frac{1}{2} \leq x \leq 1\end{array}\right.$

(e)

$y=f(x)=\cos ^{-1}\left(4 x^{3}-3 x\right)$

$=\left\{\begin{array}{ll}3 \cos ^{-1} x-2 \pi & \text { if } & -1 \leq x \leq-\frac{1}{2} \\ 2 \pi-3 \cos ^{-1} x & \text { if } & -\frac{1}{2} \leq x \leq \frac{1}{2} \\ 3 \cos ^{-1} x & \text { if } & \frac{1}{2} \leq x \leq 1\end{array}\right.$

(f)

$\sin ^{-1}\left(2 x \sqrt{1-x^{2}}\right)$

$=\left\{\begin{array}{ll}-\left(\pi+2 \sin ^{-1} x\right) & -1 \leq x \leq-\frac{1}{\sqrt{2}} \\ 2 \sin ^{-1} x & -\frac{1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}} \\ \pi-2 \sin ^{-1} x & \frac{1}{\sqrt{2}} \leq x \leq 1\end{array}\right.$

(g)

$\cos ^{-1}\left(2 x^{2}-1\right)$

$=\left\{\begin{array}{ll}2\cos ^{-1} x & 0 \leq x \leq 1 \\ 2 \pi-2 \cos ^{-1} x & -1 \leq x \leq 0\end{array}\right.$

Indefinite Integration - Notes, Concept and All Important Formula

INDEFINITE INTEGRATION If f & F are function of

$x$ such that

$F^{\prime}(x)$

$=f(x)$ then the function

$F$ is called a PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of

$\mathrm{f}(\mathrm{x})$ w.r.t.

$\mathrm{x}$ and is written symbolically as

$\displaystyle \int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}$

$=\mathrm{F}(\mathrm{x})+\mathrm{c} \Leftrightarrow \dfrac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{F}(\mathrm{x})+\mathrm{c}\}$

$=\mathrm{f}(\mathrm{x})$ , where

$\mathrm{c}$ is called the constant of integration. Note : If

$\displaystyle \int f(x) d x$

$=F(x)+c$ , then

$\displaystyle \int f(a x+b) d x$

$=\dfrac{F(a x+b)}{a}+c, a \neq 0$ All Chapter Notes, Concept and Important Formula 1. STANDARD RESULTS : (i)

$\displaystyle \int(a x+b)^{n} d x$

$=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c ; n \neq-1$ (ii)

$\displaystyle \int \dfrac{d x}{a x+b}$

$=\dfrac{1}{a} \ln|a x+b|+c$ (iii)

$\displaystyle \int e^{\mathrm{ax}+b} \mathrm{dx}$ \(=\dfrac{1}{...

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