indefinite-integration

General Integrations

• $\int adx=ax+C$

• $\int x^{n}dx=\frac{x^{n+1}}{n+1}+Cn\ne -1$

• $\int \frac{1}{x}dx=\ln |x|+C$

• $\int e^{x}dx=e^x+C$

• $\int a^{x}dx=\frac{a^x}{\ln (a)}+C$

• $\int \sin (x)dx=-\cos (x)+C$

• $\int \cos (x)dx=\sin (x)+C$

• $\int {\sec }^2(x)dx=\tan (x)+C$

• $\int {\csc }^2(x)dx=-\cot (x)+C$

• $\int \tan (x)dx=-\ln |\cos (x)|+C$

• $\int \cot (x)dx=\ln |\sin (x)|+C$

• secant integration

  • $\int \sec (x)dx=\ln |\sec (x)+\tan (x)|+C$
  • $\int \sec (x)dx=-\ln |\sec (x)-\tan (x)|+C$
  • $\int \sec (x)dx=\ln |\tan (\frac{\pi }{4}+\frac{x}{2})|+C$

• cosecant integration

  • $\int \csc (x)dx=\ln |\csc (x)-\cot (x)|+C$
  • $\int \csc (x)dx=-\ln |\csc (x)+\cot (x)|+C$
  • $\int \csc (x)dx=\ln |\tan (\frac{x}{2})|+C$

• $\int \sec (x)\tan (x)dx=\sec (x)+C$

• $\int \frac{\frac{df(x)}{dx}}{f(x)}dx=\ln (f(x))+C$

• $\int (f(x){)}^n\frac{df(x)}{dx}dx=\frac{(f(x){)}^{n+1}}{n+1}+C$

• $\int \frac{\frac{df(x)}{dx}}{\sqrt{f(x)}}dx=2\sqrt{f(x)}+C$

Special Integrations

• general inverse trigonometry substitution results
  • $\int \frac{1}{\sqrt{a^2-x^2}}dx={\sin }^{-1}(\frac{x}{a})+C$
  • $\int \frac{1}{a^2+x^2}dx=\frac{1}{a}{\tan }^{-1}(\frac{x}{a})+C$
  • $\int \frac{1}{x\sqrt{x^2-a^2}}dx=\frac{1}{a}{\sec }^{-1}(\frac{x}{a})+C$
  • $\int \frac{-1}{\sqrt{a^2-x^2}}dx={\cos }^{-1}(\frac{x}{a})+C$
  • $\int \frac{-1}{a^2+x^2}dx=\frac{1}{a}{\cot }^{-1}(\frac{x}{a})+C$
  • $\int \frac{-1}{x\sqrt{x^2-a^2}}dx=\frac{1}{a}{\csc }^{-1}(\frac{x}{a})+C$
• general logarithms substitution results
  • $\int \frac{1}{a^2-x^2}dx=\frac{1}{2a}\ln |\frac{a+x}{a-x}|+C$
  • $\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln |\frac{a-x}{a+x}|+C$
  • $\int \frac{1}{\sqrt{x^2+a^2}}dx=\ln |x+\sqrt{x^2+a^2}|+C$
  • $\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln |x+\sqrt{x^2-a^2}|+C$
• inverse and algebra combined general results
  • $\int \sqrt{a^2-x^2}dx=\frac{1}{2}x\sqrt{a^2-x^2}+\frac{1}{2}{a}^2{\sin }^{-1}(\frac{x}{a})+C$
  • $\int \sqrt{x^2+a^2}dx=\frac{1}{2}x\sqrt{x^2+a^2}+\frac{1}{2}{a}^2\ln |x+\sqrt{x^2+a^2}|+C$
  • $\int \sqrt{x^2-a^2}dx=\frac{1}{2}x\sqrt{x^2-a^2}-\frac{1}{2}{a}^2\ln |x+\sqrt{x^2-a^2}|+C$

Important Substituitions

• Whenever there is a high power inside a power in integrate, take it common, put it equal to t and apply substituion.

• If log is becoming a problem, using substitution and put that logarithm term as t

• Wherever you see $(\sqrt{x^2+1}+x)$ in integration, just put it equal to t and apply substitution.

  • $\sqrt{x^2+1}+x=\frac{1}{\sqrt{x^2+1}-x}$
  • Wherever you see $(\sqrt{x^2+1}-x)$ in integration, just put it equal to 1/t and apply substitution.

• For following integration $\int {\sin }^m(x){\cos }^n(x)dx$

  • If the index of sinx is a positive odd integer, put cosx = t and apply substituition
  • If the index of cosx is a positive odd integer, put sinx = t and apply substituition

• Never fear to put $\frac{linear}{linear}$ = t and apply substitution

• If sin(2x) has cosx - sinx near it , put t = cosx + sinx

• If sin(2x) has cosx + sinx near it , put t = cosx - sinx

• $a^2-x^2\text{ or }\sqrt{a^2-x^2}\text{put x}=a\sin \theta \text{ or x =}a\cos \theta$

• $a^2+x^2\text{ or }\sqrt{a^2+x^2}\text{put x}=a\tan \theta \text{ or x =}a\cot \theta$

• $x^2-a^2\text{ or }\sqrt{x^2-a^2}\text{put x}=a\sec \theta \text{ or x =}a\csc \theta$

• put x = $a\cos (2\theta )$ for following $\sqrt{a+x}\text{ or }\sqrt{a-x}$ $\sqrt{\frac{a+x}{a-x}}\text{ or }\sqrt{\frac{a-x}{a+x}}$

• put x = $a{\cos }^2(\theta )+b{\sin }^2(\theta )$ for following $\sqrt{\frac{x-a}{b-x}}\text{ or }\sqrt{(x-a)(b-x)}$

Product Rule Results

• $\int e^x(f(x)+f^{\prime }(x))dx=e^{x}f(x)+C$
• $\int (x{f}^{\prime }(x)+f(x))dx=xf(x)+C$
• $\int e^{ax}cosbxdx=\frac{e^{ax}}{(a^2+b^2)}(acosbx+bsinbx)+C$
• $\int e^{ax}sinbxdx=\frac{e^{ax}}{a^2+b^2}(asinbx-bcosbx)+C$
• $\int e^{g(x)}(f(x).g^{\prime }(x)+f^{\prime }(x))dx=e^{g(x)}f(x)+C$
• $\int \ln (x)dx=x\ln (x)-x+C$
• $\int \ln (x)dx=x\ln (x)-x+C$
• $\int e^{x^2}dx=e^{x^2}{\sum }_{k=0}^{\infty }(-1{)}^k\frac{d^k(x^2{)}^k}{d{x}^k}=e^{x^2}\left[1-\frac{d(x^2)}{dx}+\frac{d^2(x^2{)}^2}{d{x}^2}-\frac{d^3(x^2{)}^3}{d{x}^3}+...\right]$

Integration of Algebric Expressions

$\int \frac{P(x)}{a{x}^2+bx+c}\text{ or }\int \frac{P(x)}{\sqrt{a{x}^2+bx+c}}$

• if P(x) is constant, just complete the square in the denominator
• if P(x) is linear or quadratic then use the algorithm to represent quadratic/linear in form of another
• if degree of P(x) >= 2 then you can also use the devision algorithm to write P(x) in terms of $a{x}^2+bx+c$

  IMPORTANT

  Above point can be used for any algebric expression where degree of numerator is greater than or equal to degree of denominator

$\int \frac{x^2+a^2}{x^4+K{x}^2+a^4}dx\int \frac{x^2-a^2}{x^4+K{x}^2+a^4}dx\int \frac{dx}{x^4+K{x}^2+a^4}dx\int \frac{x^{2}dx}{x^4+K{x}^2+a^4}dx$ For above integrals, devide both numerator and denominator by $x^2$

$1-\frac{1}{x^2}=\frac{d}{dx}(x+\frac{1}{x})$ $1+\frac{1}{x^2}=\frac{d}{dx}(x-\frac{1}{x})$ $x^2+\frac{1}{x^2}=(x+\frac{1}{x}{)}^2-2$ $x^2+\frac{1}{x^2}=(x-\frac{1}{x}{)}^2+2$

• If you want to represent a linear equation in form of derivative of another quadratic, you can use following formula $ax+b=\alpha (\frac{d(p{x}^2+qx+r)}{dx})+\beta$

After simplifying above equations, just compare the coefficients of various terms for find out value of the contstants alpha, beta and lambda

Integration by Partial Fractions

$\frac{P(x)}{(x-a_1)(x-a_2)}=\frac{A}{x-a_1}+\frac{B}{x-a_2}$ $\frac{P(x)}{(x-a_1{)}^2(x-a_2)}=\frac{A}{x-a_1}+\frac{B}{(x-a_1{)}^2}+\frac{C}{x-a_2}$ $\frac{P(x)}{(a{x}^2+bx+c)(x-a_1)}=\frac{Ax+B}{a{x}^2+bx+c}+\frac{C}{x-a_1}$ $\frac{P(x)}{(a{x}^2+bx+c{)}^2(x-a_1{)}^2}=\frac{Ax+B}{a{x}^2+bx+c}+\frac{Cx+D}{(a{x}^2+bx+c{)}^2}+\frac{E}{x-a_1}+\frac{F}{(x-a_1{)}^2}$

Integration By Parts

$\int f(x)g(x)dx=f(x)\int g(x)dx-\int [\frac{df(x)}{dx}\times \int g(x)dx]dx$

• Function whose integration is easy, is taken as the second function
• If integrad contains only 1 function which cannot be integrated directory, then we take second function as 1 and try By Parts
• Usually, first function is taken in order of ILATE i.e inverse function, log function, algebric function, trigonometric function and then exponential function