Integration

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

  1. \(\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{b}{f(t)dt}\)
  2. \(\int\limits_{a}^{b}{f(x)dx=}-\int\limits_{b}^{a}{f(x)dx}\)
  3. \(\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{c}{f(x)dx}+\int\limits_{c}^{b}{f(x)dx}\)
  4. \(\int\limits_{a}^{b}{f(x)dx=}\int\limits_{a}^{b}{f(a+b-x)dx}\)
  5. \(\int\limits_{0}^{2a}{f(x)dx}=\int\limits_{0}^{a}{f(x)dx}+\int\limits_{0}^{a}{f(2a-x)dx}\)\(=0\,\,if\,\,f(2a-x)=-f(x)\) and \(=2\int\limits_{0}^{a}{f(x)}\,\,if\,\,f(2a-x)=f(x)\)
  6. \(\int\limits_{-a}^{a}{f(x)dx=\left\{ \begin{matrix} 0 & if\,\,f\left( x \right)\,is\,odd \\ 2\int\limits_{0}^{a}{f\left( x \right)dx} & if\,\,f\left( x \right)\,is\,odd \\ \end{matrix} \right.\,\,\,\,\,\,}\)

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 odd

Area

\(A=\int\limits_{a}^{b}{f(x)dx=\int\limits_{a}^{b}{y\,dx}}\)

\(A=\int\limits_{a}^{b}{\left( f(x)-g(x) \right)}\,dx\)

Illustration:

Find area between \({{y}^{2}}=12x\)and \({{x}^{2}}=12y\)

\(A=\int\limits_{0}^{12}{\sqrt{12x}-\frac{{{x}^{2}}}{12}dx}\) \(=\frac{12{{x}^{{3}/{2}\;}}}{{3}/{2}\;}-\frac{{{x}^{2}}}{12}dx\) \(=\left. 8{{x}^{{3}/{2}\;}} \right|_{0}^{12}-\left. \frac{{{x}^{3}}}{36} \right|_{0}^{12}\) \(=8\,\,.\,\,{{12}^{{3}/{2}\;}}-\frac{144}{3}\)

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:

\(A=\int\limits_{0}^{{{\left( \sqrt{3}+1 \right)}^{2}}}{\sqrt{4x}\,\,dx-}\)Area (semicircle) – Area of ΔABC

\(=\sqrt{4}\left[ \frac{2{{x}^{3/2}}}{3} \right]_{0}^{{{\left( \sqrt{3}+1 \right)}^{2}}}-\frac{\pi }{2}-\frac{1}{2}{{\left( \sqrt{3}+1 \right)}^{2}}2\left( \sqrt{3}+1 \right)\) \(=\frac{{{\left( \sqrt{3}+1 \right)}^{3}}}{3}-\frac{\pi }{2}\)

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

\(3{{\alpha }^{2}}-8\alpha ={{\alpha }^{2}}+2-f\left( \alpha \right)\) \(f\left( x \right)=-2{{x}^{2}}+8x+2\)

 


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etan1x(1+x+x2).d(cot1x) is equal to