Integrand form | Substitution | |
(i) | \[\sqrt{{{a}^{2}}-{{x}^{2}}},\frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}},{{a}^{2}}-{{x}^{2}}\] | \[x=a\sin \theta ,\] or \[x=a\cos \theta \] |
(ii) | \[\sqrt{{{x}^{2}}+{{a}^{2}}},\,\frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}},\,{{x}^{2}}+{{a}^{2}}\] | \[x=a\tan \theta \] or \[x=a\sin \text{h}\theta \] |
(iii) | \[\sqrt{{{x}^{2}}-{{a}^{2}},}\,\,\frac{1}{\sqrt{{{x}^{2}}-{{a}^{2}}}},\,\,{{x}^{2}}-{{a}^{2}}\] | \[x=a\sec \theta \] or \[x=a\cosh \theta \] |
(iv) | \[\sqrt{\frac{x}{a+x}},\,\,\sqrt{\frac{a+x}{x}},\,\,\sqrt{x(a+x)},\,\,\frac{1}{\sqrt{x(a+x)}}\] | \[x=a{{\tan }^{2}}\theta \] |
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Two equal unlike parallel forces which do not have the same line of action, are said to form a couple.
Example : Couples have to be applied in order to wind a watch, to drive a gimlet, to push a cork screw in a cork or to draw circles by means of pair of compasses.
(1) Arm of the couple : The perpendicular distance between the lines of action of the forces forming the couple is known as the arm of the couple.
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The moment of a force about a point O is given in magnitude by the product of the forces and the perpendicular distance of O from the line of action of the force.
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(1) Like parallel forces : Two parallel forces are said to be like parallel forces when they act in the same direction.
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If three forces acting at a point be in equilibrium, each force is proportional to the sine of the angle between the other two. Thus if the forces are \[P,\,\,Q\] and \[R;\] \[\alpha ,\,\,\beta ,\,\,\gamma \] be the angles between \[Q\] and \[R,\,\,R\] and \[P,\,\,P\] and \[Q\] respectively, also the forces are in equilibrium, we have, \[\frac{P}{\sin \alpha }=\frac{Q}{\sin \beta }=\frac{R}{\sin \gamma }\].
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If any number of forces acting on a particle be represented in magnitude and direction by the sides of a polygon taken in order, the forces shall be in equilibrium.
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If three forces, acting at a point, be represented in magnitude and direction by the sides of a triangle, taken in order, they will be in equilibrium.
Here \[\overrightarrow{AB}=P,\ \ \ \overrightarrow{BC}=Q,\ \ \overrightarrow{CA}=R\]
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If two forces, acting at a point, be represented in magnitude and direction by the two sides of a parallelogram drawn from one of its angular points, their resultant is represented both in magnitude and direction of the parallelogram drawn through that point.
If \[OA\] and \[OB\] represent the forces \[P\] and \[Q\] acting at a point \[O\] and inclined to each other at an angle\[\alpha \]. If \[R\] is the resultant of these forces represented by the diagonal \[OC\] of the parallelogram \[OACB\] and \[R\] makes an angle \[\theta \] with \[P\].
i.e., \[\angle COA=\theta \], then \[{{R}^{2}}={{P}^{2}}+{{Q}^{2}}+2PQ\cos \alpha \] and \[\tan \theta =\frac{Q\sin \alpha }{P+Q\cos \alpha }\]
The angle \[{{\theta }_{1}}\] which the resultant \[R\] makes with the direction of the force \[Q\] is given by \[{{\theta }_{1}}={{\tan }^{-1}}\left( \frac{P\sin \alpha }{Q+P\cos \alpha } \right)\]
Case (i) : If \[P=Q\]
\[\therefore R=2P\cos \,\left( \frac{\alpha }{2} \right)\] and \[\tan \theta =\tan \left( \frac{\alpha }{2} \right)\] or \[\theta =\frac{\alpha }{2}\]
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(1) (i) \[\int{{{x}^{n}}dx=\frac{{{x}^{n+1}}}{n+1}+c,\,n\ne -1}\]
(ii) \[\int{{{(ax+b)}^{n}}dx=\frac{1}{a}.\,\frac{{{(ax+b)}^{n+1}}}{n+1}}+c\], \[n\ne -1\]
(2) (i) \[\int{\frac{1}{x}dx=\log |x|+c}\]
(ii) \[\int{\frac{1}{ax+b}\,dx=\frac{1}{a}(\log |ax+b|)+c}\]
(3) \[\int{{{e}^{x}}dx={{e}^{x}}+c}\]
(4) \[\int{{{a}^{x}}dx=\frac{{{a}^{x}}}{{{\log }_{e}}a}+c}\]
(5) \[\int{\sin x\,dx=-\cos x+c}\]
(6) \[\int{\cos x\,dx=\sin x+c}\]
(7) \[\int{{{\sec }^{2}}}x\,dx=\tan x+c\]
(8) \[\int{\text{cos}\text{e}{{\text{c}}^{2}}x\,dx}=-\cot x+c\]
(9) \[\int{\sec x\,\tan x\,dx=\sec x+c}\]
(10) \[\int{\text{cosec}\,x\,\cot x\,dx=-\text{cosec}\,x+c}\]
(11) \[\int{\tan x\,dx}=-\log |\cos x|+c=\log |\sec x|+c\]
(12) \[\int{\cot x\,dx=\log |\sin x|+}c=-\log |\cos ec\,x|+c\]
(13) \[\int{\sec x\,dx=\log |\sec x+\tan x|+c=\log \tan \left( \frac{\pi }{4}+\frac{x}{2} \right)+c}\]
(14) \[\int{\text{cos}\text{ec}}\,x\,dx=\log |\text{cos}\text{ec}\,x-\cot x|+c=\log \tan \frac{x}{2}+c\]
(15) \[\int{\frac{dx}{\sqrt{1-{{x}^{2}}}}={{\sin }^{-1}}x+c=-{{\cos }^{-1}}x+c}\]
(16) \[\int{\frac{dx}{\sqrt{{{a}^{2}}-{{x}^{2}}}}={{\sin }^{-1}}\frac{x}{a}+c}=-{{\cos }^{-1}}\frac{x}{a}+c\]
(17) \[\int{\frac{dx}{1+{{x}^{2}}}={{\tan }^{-1}}x+c=-{{\cot }^{-1}}x+c}\]
(18) \[\int{\frac{dx}{{{a}^{2}}+{{x}^{2}}}=\frac{1}{a}{{\tan }^{-1}}\frac{x}{a}+c=\frac{-1}{a}{{\cot }^{-1}}\frac{x}{a}+c}\]
(19) \[\int{\frac{dx}{x\sqrt{{{x}^{2}}-1}}={{\sec }^{-1}}x+c}=-\cos e{{c}^{-1}}x+c\]
(20) \[\int{\frac{dx}{x\sqrt{{{x}^{2}}-{{a}^{2}}}}=\frac{1}{a}{{\sec }^{-1}}\frac{x}{a}+c}=\frac{-1}{a}\cos e{{c}^{-1}}\frac{x}{a}+c\]
In any of the fundamental integration formulae, if \[x\] is replaced by \[ax+b\], then the same formulae is applicable but we must divide by coefficient of \[x\] or derivative of \[(ax+b)\] i.e., \[a\]. In general, if \[\int{f(x)dx=\varphi (x)+c}\], then\[\int{f(ax+b)\,dx=\frac{1}{a}\varphi \,(ax+b)}+c\]
\[\int{\sin (ax+b)\,dx=\frac{-1}{a}}\cos (ax+b)+c,\]
\[\int{\sec (ax+b)\,dx=\frac{1}{a}\log |\sec (ax+b)+\tan (ax+b)|+c}\] etc.
Some more results :
(i) \[\int{\frac{1}{{{x}^{2}}-{{a}^{2}}}=\frac{1}{2a}\log \left| \frac{x-a}{x+a} \right|+c=\frac{-1}{a}{{\coth }^{-1}}\frac{x}{a}+c}\], when \[x>a\]
(ii) \[\int{\frac{1}{{{a}^{2}}-{{x}^{2}}}dx=\frac{1}{2a}\log \left| \frac{a+x}{a-x} \right|+c=\frac{1}{a}{{\tanh }^{-1}}\frac{x}{a}+c}\], when \[x<a\]
(iii) \[\int{\frac{dx}{\sqrt{{{x}^{2}}-{{a}^{2}}}}=\log \{|x+\sqrt{{{x}^{2}}-{{a}^{2}}}|\}+c=\cos \,{{\text{h}}^{-1}}\left( \frac{x}{a} \right)}+c\]
(iv) \[\int{\frac{dx}{\sqrt{{{x}^{2}}+{{a}^{2}}}}=\log }\{|x+\sqrt{{{x}^{2}}+{{a}^{2}}}|\}+c=\sin {{\text{h}}^{-1}}\left( \frac{x}{a} \right)+c\]
(v) \[\int{\sqrt{{{a}^{2}}-{{x}^{2}}}dx=\frac{1}{2}x\sqrt{{{a}^{2}}-{{x}^{2}}}+\frac{1}{2}{{a}^{2}}{{\sin }^{-1}}\left( \frac{x}{a} \right)+c}\]
(vi) \[\int_{{}}^{{}}{\sqrt{{{x}^{2}}-{{a}^{2}}}}dx=\frac{1}{2}x\sqrt{{{x}^{2}}-{{a}^{2}}}-\frac{1}{2}{{a}^{2}}\log \{x+\sqrt{{{x}^{2}}-{{a}^{2}}}\}+c\]\[=\frac{1}{2}x\sqrt{{{x}^{2}}-{{a}^{2}}}-\frac{1}{2}{{a}^{2}}{{\cosh }^{-1}}\left( \frac{x}{a} \right)+c\]
(vii) \[\int{\sqrt{{{x}^{2}}+{{a}^{2}}}dx=\frac{1}{2}x\sqrt{{{x}^{2}}+{{a}^{2}}}+\frac{1}{2}{{a}^{2}}\log \{x+\sqrt{{{x}^{2}}+{{a}^{2}}}\}+c}\]\[=\frac{1}{2}x\sqrt{{{x}^{2}}+{{a}^{2}}}+\frac{1}{2}{{a}^{2}}\sin {{\text{h}}^{-1}}\left( \frac{x}{a} \right)\]
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