Properties of Integration
Property 1: \(\int\limits_{a}^{a}{f(x)\,dx=0}\)
Property 2: \(\int\limits_{a}^{b}{f(x)=}-\int\limits_{b}^{a}{f(x)dx}\)
Property 3: \(\int\limits_{a}^{b}{f(x)=}\int\limits_{a}^{b}{f(t)}\)
Property 4: \(\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{c}{f(x)dx+\int\limits_{a}^{b}{f(x)dx}}\)
Property 5: (i) \(\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{b}{f(a+b-x)dx}\)
(ii) \(\int\limits_{0}^{a}{f(x)dx=}\int\limits_{0}^{a}{f(a-x)dx}\)
Q \(f(x)=\left\{ \begin{matrix} {{x}^{2}}+x & 1\le x\le 2 \\ 3x & 2\le x\le 3 \\ \end{matrix} \right.\) \(\int\limits_{1}^{3}{f(x)=\frac{34}{3}}\)
Q \(\int\limits_{-1}^{1}{{{e}^{(x)}}}dx=2(e-1)\)
Q \(\int\limits_{-2}^{3}{\left| \left. x \right| \right.dx=\frac{13}{2}}\) \(\int\limits_{-1}^{1}{{{x}^{3}}\left| x \right|dx=0}\)
Q \(\int\limits_{2}^{8}{\left| x-5 \right|dx=9}\) \(\int\limits_{{}^{1}/{}_{4}}^{1}{\left| 2x-1 \right|dx=\frac{5}{16}}\)
Q \(\int\limits_{{}^{\pi }/{}_{2}}^{{}^{\pi }/{}_{2}}{\left| \sin x \right|dx=2}\) \(\int\limits_{-2}^{2}{\left| x \right|+\left| x-1 \right|dx=9}\)
⇒ Also Read Definite and Indefinite Integration
Useful Formulas
- \(\int{{{e}^{ax}}\sin bx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}\left[ a\sin bx-b\cos bx \right]}\)
- \(\int{{{e}^{ax}}\cos bx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}\left[ a\cos bx+b\sin bx \right]}\)
Illustration:
- \(\int{{{e}^{-2x}}\sin x\cos 3xdx}\)
- \(=\int{{{e}^{-2x}}\left( \sin 4x-\sin 2x \right)dx}\)
- \(={{e}^{-2x}}\left[ \left( \frac{\sin 2x+\cos 2x}{8} \right)-\frac{\sin 4x+2\cos 4x}{20} \right]\)
- \(\int{{{e}^{x}}\left( f(x)+f'(x) \right)}={{e}^{x}}f(x)\)
Illustration:
\(\int{{{e}^{x}}\left( \frac{1}{x}+\frac{1}{{{x}^{2}}} \right)dx}={{\frac{e}{x}}^{x}}+c\) \(\int{{{e}^{x}}(\sin x+\cos x)dx={{e}^{x}}\sin x+c}\) \(\int{{{e}^{x}}(lnx+\frac{1}{x})dx={{e}^{x}}lnx+c}\)Integration of Trigonometric Functions
Type 1: \(I=\int{{{\sin }^{m}}x{{\cos }^{n}}xdx}\)
1. If m –odd put \(\cos x=t\)
2. If n odd put \(\sin x=t\)
3. If m, n rationales then put \(\tan x=t\)
4. If both even then use reduction method
\(Q\int{\frac{{{\cos }^{3}}x}{{{\sin }^{6}}x}dx=\int{\frac{1-{{t}^{2}}}{{{t}^{6}}}dt}}\)Where \(t=\sin x\) \(=\int{{{t}^{-6}}-{{t}^{-4}}dt}\) \(=-\frac{1}{5si{{n}^{5}}x}+\frac{1}{3{{\sin }^{3}}x}+c\)
Type 2: \(\int{\frac{dx}{a\cos x+b\sin x+c}}\)
Put \(t=\tan \left( \frac{x}{2} \right)\)
Illustration
\(\int{\frac{dx}{2+\sin x}}\) \(\Rightarrow t=\tan \left( \frac{x}{2} \right)\) \(dx=\frac{2dt}{1+{{t}^{2}}}\) \(=\int{\frac{\frac{2dt}{1+{{t}^{2}}}}{2+\frac{2t}{1+{{t}^{2}}}}}\) \(\Rightarrow \int{\frac{dt}{{{t}^{2}}+t+1}}\) \(=\frac{2}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{2t+1}{\sqrt{3}} \right)\) \(=\frac{2}{\sqrt{3}}{{\tan }^{-1}}\left( \frac{2\tan \frac{x}{2}+1}{\sqrt{3}} \right)+c\)Some useful Substitutions for Irrational Functions
- Form 1 : \(\int{linear\sqrt{Quadralis}\,dx}\)
Substitute \(liner=mQudratic’+n\)
- Form 2: \(\int{\frac{dx}{lin\sqrt{li{{n}_{1}}}},}\,\int{\frac{lin}{\sqrt{li{{n}_{1}}}}}dx,\,\int{\frac{\sqrt{li{{n}_{1}}}}{lin}dx}\)
Substitute \(li{{n}_{1}}={{t}^{2}}\)
- Form 3: \(\int{\frac{1}{lin\sqrt{Qua}}dx}\)
Substitute \(lin=\frac{1}{t}\)
- Form 4: \(\int{\frac{dx}{\left( a{{x}^{2}}+b \right)\sqrt{\left( {{x}^{2}}+d \right)}}}\)
Substitute \(x=\frac{1}{t}\)& then, \({{u}^{2}}\)for \(a{{t}^{2}}+b\)
Integration Formulas
- \(\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{b}{f(t)dt}\)
- \(\int\limits_{a}^{b}{f(x)dx=}-\int\limits_{b}^{a}{f(x)dx}\)
- \(\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{c}{f(x)dx}+\int\limits_{c}^{b}{f(x)dx}\)
- \(\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{b}{f(a+b-x)dx}\)
- \(\int\limits_{0}^{2a}{f(x)dx}=\int\limits_{0}^{a}{f(x)dx}+\int\limits_{0}^{a}{f(2a-x)dx}\)\(=0\,\,if\,\,f(2a-x)=-f(x)\) and \(=2\int\limits_{0}^{a}{f(x)}\,\,if\,\,f(2a-x)=f(x)\)
- \(\int\limits_{-a}^{a}{f(x)dx=\left\{ \begin{matrix} 0 & if\,\,f\left( x \right)\,is\,odd \\ 2\int\limits_{0}^{a}{f\left( x \right)dx} & if\,\,f\left( x \right)\,is\,odd \\ \end{matrix} \right.\,\,\,\,\,\,}\)
Problems on Integration
Illustration:
\(\int\limits_{0}^{2}{{{x}^{2}}\left[ x \right]dx=}\int\limits_{0}^{1}{{{x}^{2}}\left[ x \right]dx}+\int\limits_{1}^{2}{{{x}^{2}}\left[ x \right]dx}\) \(=\int\limits_{0}^{1}{{{x}^{2}}.0\,dx}+\int\limits_{1}^{2}{{{x}^{2}}\left[ 1 \right]dx}\) \(=0+\left. \frac{{{x}^{3}}}{3} \right|_{1}^{2}\) \(=\frac{8-1}{3}\,=\frac{7}{3}\)Illustration:
\(\int\limits_{{\pi }/{6}\;}^{{\pi }/{3}\;}{\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx=}\int{\frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx}\) (by \(x\to \frac{\pi }{2}-x\)) \(2I=\int\limits_{{\pi }/{6}\;}^{{\pi }/{3}\;}{\frac{\sqrt{\sin x}+\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}}=\int\limits_{{\pi }/{6}\;}^{{\pi }/{3}\;}{1\,dx}=\frac{\pi }{6}\) \(I=\frac{\pi }{12}\)Illustration:
\(I=\int{{{\sin }^{100}}x{{\cos }^{99}}x}\)Here \(f(2\pi -x)=f(x)\)
Or \(I=2\int\limits_{0}^{\pi }{{{\sin }^{100}}x{{\cos }^{99}}x}\) \(=2\int\limits_{0}^{\pi }{{{\sin }^{100}}\left( \pi -x \right){{\cos }^{99}}\left( \pi -x \right)}\)
I = -I
I = 0
Illustration:
\(\int\limits_{-5}^{5}{{{x}^{3}}=0}\) as \(f(x)\) is oddArea
Illustration:
Find area between \({{y}^{2}}=12x\)and \({{x}^{2}}=12y\)
Linear Differential Equation
Equation of form \(\frac{dy}{dx}+py=\theta\).
Ar solve by multiplying integrating function
If \({{e}^{\int{pdx}}}\)
To get final solution of
\(y(if)=\int{Q(IF)+c}\)Illustration
Solve \({{x}^{2}}\frac{dy}{dx+y+1}=1\)
Ans : Given equation is
\(\frac{dy}{dx}+\frac{1}{{{x}^{2}}}y=\frac{1x}{{{x}^{2}}}\)which is linear \(p=\frac{1}{{{x}^{2}}},q=\frac{1}{{{x}^{2}}}\)IF\(={{e}^{\int{\frac{1}{{{x}^{x}}}dx}}}=e\frac{1}{x}\)
Ans:
\(y\,e\,\frac{-1}{x}=\int{{{p}^{\frac{-1}{{{x}^{2}}}}}}(\frac{1}{{{x}^{2}}})dx+c\) \(y=1+c{{e}^{\frac{1}{x}}}\)General Form
We use combinations of different terms to get united term
Following terms are useful
1. \(xdy+ydx=d(xy)\)
2. \(\frac{xdy-ydx}{{{x}^{2}}}=d\left( \frac{y}{x} \right)\).
3. \(\frac{ydx-xdy}{{{y}^{2}}}=d\left( \frac{x}{y} \right)\)
4. \(\frac{xdy+ydx}{xy}=d(la\,xy)\)
5. \(\frac{xdy-ydx}{xy}=d\left( 1\left( \frac{y}{x} \right) \right)\)
6. \(\frac{xdy-ydx}{{{x}^{2}}+{{y}^{2}}}=d\left( {{\tan }^{-1}}\left( \frac{y}{x} \right) \right)\)
Illustration
Solve \(xdx+ydy=\frac{xdy-ydx}{{{x}^{2}}+{{y}^{2}}}\)
Ans:
\(\frac{1}{2}d({{x}^{2}}+{{y}^{2}})=d\left( {{\tan }^{-1}}\left( \frac{y}{x} \right) \right)\)Integrating we get
\(\frac{1}{2}({{x}^{2}}+{{y}^{2}})={{\tan }^{-1}}\left( \frac{y}{x} \right)+c\)Leibnitz’s rule
f cont [a, b] & \(u\left( x \right),v\left( x \right)\) different function.
\(\frac{d}{dx}\int\limits_{u\left( x \right)}^{v\left( x \right)}{f\left( t \right)dt=f\left( v\left( x \right) \right)}\frac{dv\left( x \right)}{dx}-f\left( u\left( x \right) \right)u’\left( x \right)\)Proof:
\(\frac{d}{dx}F\left( x \right)=f\left( x \right)\)∴ \(\int\limits_{u\left( x \right)}^{v\left( x \right)}{f\left( t \right)dt=F\left( v\left( x \right) \right)}-F\left( u\left( x \right) \right)\)
Practice Problems
Q. \(\int\limits_{{{x}^{2}}}^{{{x}^{3}}}{\frac{1}{\log t}dt=y}\) find \(\frac{dy}{dx}=x\left( x-1 \right){{\left( \log x \right)}^{-1}}\)
Q. If \(\int\limits_{\sin x}^{1}{{{t}^{2}}f\left( t \right)dt=1-\sin x.}\) where \(x\in \left( 0,\frac{\pi }{2} \right)\) . find \(f\left( \frac{1}{\sqrt{3}} \right)=3\)
Q. \(f\left( 2 \right)=6,f’\left( 2 \right)=\frac{1}{48}\). Find \(\underset{x\to 2}{\mathop{\lim }}\,\int\limits_{6}^{f\left( x \right)}{\frac{4{{t}^{3}}}{x-2}}dt=18\)
Q. \(\underset{x\to \infty }{\mathop{\lim }}\,\frac{\int\limits_{0}^{x}{{{e}^{{{x}^{2}}}}dx}}{\int\limits_{0}^{x}{{{e}^{2{{x}^{2}}}}dx}}=0\)
Q. \(y=\int\limits_{\frac{1}{8}}^{{{\sin }^{2}}x}{{{\sin }^{-1}}\sqrt{t}\,dt}+\int\limits_{\frac{1}{8}}^{{{\cos }^{2}}x}{{{\cos }^{-1}}\sqrt{t}\,dt}.\) \(x\in \left[ 0,\frac{\pi }{2} \right]\) is equation of straight line. Find it
∴ \(\left( \frac{dy}{dx}=0 \right)y=const\) put \(x=\frac{\pi }{4}y=\frac{3\pi }{16}\)
Q. \(f:\left( 0,\infty \right)\to \left( 0,\infty \right)\) \(x\int\limits_{0}^{x}{\left( 1-t \right)f\left( t \right)dt=\int\limits_{0}^{x}{t\,f\left( t \right)dt}}\,\,\,\,\,\,x\in {{R}^{+}}\)
Given \(f\left( 1 \right)=1.\) Find \(f\left( x \right).\,\,\,\,\,\,\,f\left( x \right)=\frac{1}{{{x}^{3}}}{{e}^{\left( 1-\frac{1}{x} \right)}}\)
Hint differentiate twice. \(\int{\frac{f’x}{f\left( x \right)}}=\int{\frac{1-3x}{{{x}^{2}}}}\)
Q. \(\underset{x\to 4}{\mathop{\lim }}\,\int\limits_{4}^{x}{\frac{4t-f\left( t \right)}{x-4}dt=16-f\left( 4 \right)}\)
Q. \(\underset{x\to 0}{\mathop{\lim }}\,\frac{\int\limits_{0}^{x}{\cos {{t}^{2}}dt}}{x}=1\)
Q. Find points of minima for \(f\left( x \right)=\int\limits_{0}^{x}{t\left( t-1 \right)\left( t-2 \right)dt}\) Ans \(n=0,2\)
Type
\(\int{\frac{a\cos x+b\sin x+p}{c\cos x+d\sin x+q}}\,dx,\int{\frac{a{{e}^{x}}+b{{e}^{-x}}+c}{d{{e}^{x}}+f{{e}^{-x}}+h}}\,dx\)This is solved by \(Nr=nDr+mDr'\)
Illustration:
\(\int{\frac{2\cos x+3\sin x}{4\cos x+5\sin x}}\,dx\)Let \(2\cos x+3\sin x=a\left( 4\cos x+5\sin x \right)+b\left( -4\sin x+5\cos x \right)\)
Solving by comparing we get
\(a=\frac{23}{41}\,\,\,\,\,\,b=\frac{-2}{41}\)∴ \(I=\int{\frac{23}{41}-\frac{2}{41}\left( \frac{-4\sin x+5\cos x}{4\cos x+5\sin x} \right)dx}\) \(=\frac{23}{41}x-\frac{2}{41}\ell n\left| 4\cos x+5\sin x \right|+c\)
By Parts Integration
\(\int{uv\,dx}=u\int{vdx}-\int{u’\left( \int{vdx} \right)}\,dx\)Illustration:
Q. \(\int{\ell n\,x}=\int{\ell n\,\,x.1\,\,dx}\) \(=x\,\ell n\,x-\int{\frac{1}{x}.\,x}\,dx\) \(=x\,\ell n\,x-x\)
Q. \(\int{x\,{{e}^{x}}dx=x\int{{{e}^{x}}dx-\int{{{\left( x \right)}^{1}}\left( \int{{{e}^{x}}dx} \right)}dx}}\) \(=x{{e}^{x}}-\int{{{e}^{x}}dx}\) \(=x{{e}^{x}}-{{e}^{x}}\)
Q. \(\int{\left( {{x}^{3}}+3x \right)\sin 3x\,dx=\left( {{x}^{3}}+3x \right)\left( \frac{-\cos 3x}{3} \right)}-\left( 3{{x}^{2}}+3 \right)\left( \frac{-{{\sin }^{3}}x}{{{3}^{2}}} \right)+6x\left( \frac{\cos 3x}{{{3}^{3}}} \right)-6\left( \frac{{{\sin }^{3}}x}{{{3}^{4}}} \right)+c\)
Integration of Irrational Algebraic Functions
Type \(\int{\frac{dx}{{{\left( ax+b \right)}^{k}}\sqrt{px+q}}}\)
Q. \(\int{\frac{x}{\left( x-3 \right)\sqrt{x+1}}dx}\)
Put \(x+1={{t}^{2}},\) we get
\(I=\int{\frac{\left( {{t}^{2}}-1 \right)2t\,dt}{\left( {{t}^{2}}-4 \right)t}}=2\int{\frac{{{t}^{2}}-1}{{{t}^{2}}-4}dt}\) \(=2\int{1}+\frac{3}{{{t}^{2}}-4}dt\) \(=2t+\frac{3}{2}\ell n\left| \frac{t-2}{t+2} \right|+c\) \(=2\sqrt{x+1}+\frac{3}{2}\ell n\left| \frac{\sqrt{x+1}-2}{\sqrt{x+1}+2} \right|+c\)⇒ \(\int\limits_{0}^{2a}{f\left( x \right)dx=\int\limits_{0}^{a}{f\left( x \right)dx+}}\int\limits_{0}^{a}{f\left( 2a-x \right)dx}\)
= 0 if \(f\left( 2a-x \right)=-f\left( x \right)\) \(=2\int\limits_{0}^{a}{f\left( x \right)}\) if \(f\left( 2a-x \right)=f\left( x \right)\)
Illustration:
Q. \(I=\int{{{\sin }^{100}}x{{\cos }^{99}}x}\)
Here \(f\left( 2\pi -x \right)=f\left( x \right)\)
Or \(I=2\int\limits_{0}^{\pi }{{{\sin }^{100}}x{{\cos }^{99}}x}\) \(=2\int\limits_{0}^{\pi }{{{\sin }^{100}}\left( \pi -x \right)}{{\cos }^{99}}\left( \pi -x \right)\)
I = -I
I = 0
⇒ \(\int\limits_{-a}^{a}{f\left( x \right)dx=\left\{ \begin{matrix} 0 & if\,\,f\left( x \right)\,\,is\,\,odd \\ 2\int\limits_{0}^{a}{f\left( x \right)dx} & if\,\,f\left( x \right)\,\,is\,\,even \\ \end{matrix} \right.}\)
Illustration:
Q. \(\int\limits_{-5}^{5}{{{x}^{3}}=0}\) as \(f\left( x \right)\) is odd
Q. Find area between \({{y}^{2}}\le 4x,{{x}^{2}}+{{y}^{2}}\ge 2x\) and \(x\le y+2.\) in first quadrant.
Answer:
Optimisation of area for greatest and least values
Illustration:
If area by \(y=f\left( x \right)\) and \(y={{x}^{2}}+2\) between abscissa \(x=2\And x=\alpha\) is \({{\alpha }^{3}}-4{{\alpha }^{2}}+8.\) find \(f\left( x \right).\)
Answer:
\({{\alpha }^{3}}-4{{\alpha }^{2}}+8=\int\limits_{2}^{\alpha }{\left( {{x}^{2}}+2-f\left( x \right) \right)dx}\)Differentiating with Labniz equation
\(3{{\alpha }^{2}}-8\alpha ={{\alpha }^{2}}+2-f\left( \alpha \right)\) \(f\left( x \right)=-2{{x}^{2}}+8x+2\)
