Twelve Standard >> List of some frequently used formula in integration | Part-2

List of some frequently used formula in integration

1) \(\int_{}^{} sin(ax+b)dx\)=\(-\frac{1}{a}cos(ax+b)+c\)

Substituting a = 3, and b = 2, we get,
    \(\int_{}^{} sin(3x+2)dx\)=\(-\frac{1}{3}cos(3x+2)+c\)

2) \(\int_{}^{} cos(ax+b)dx\)=\(\frac{1}{a}sin(ax+b)+c\)

Substituting a = 2, and b = 5, we get,
    \(\int_{}^{} cos(2x-5)dx\)=\(\frac{1}{2}cos(2x-5)+c\)

3) \(\int_{}^{}sec^{2}(ax+b)dx\)=\(\frac{1}{a}tan(ax+b)+c\)

Substituting a = 5, and b = 8, we get,
    \(\int_{}^{}sec^{2}(5x+8)dx\)=\(\frac{1}{5}cos(5x+8)+c\)

4) \(\int_{}^{}cosec^{2}(ax+b)dx\)=\(-\frac{1}{a}cot(ax+b)+c\)

Substituting a = -5, and b = 3, we get,
   \(\int_{}^{}cosec^{2}(-5x+3)dx\)=\(-\frac{1}{-5}cot(-5x+3)+c\)
 \(\Rightarrow\)   \(\int_{}^{}cosec^{2}(-5x+3)dx\)=\(\frac{1}{5}cot(-5x+3)+c\)

5) \(\int_{}^{}tan(ax+b)dx\)=\(-\frac{1}{a}\log_{}{|cos(ax+b)|}+c\)
                      =\(\frac{1}{a}\log_{}{|sec(ax+b)|}+c\)

Substituting a = 2, and b = 3, we get,
  \(\int_{}^{}tan(2x+3)dx\)=\(\frac{1}{2}\log_{}{|cos(2x+3)|}+c\)

6) \(\int_{}^{}cot(ax+b)dx\)=\(\frac{1}{a}\log_{}{|sin(ax+b)|}+c\)
                      

Substituting a = 5, and b = 2, we get,
  \(\int_{}^{}cot(5x+2)dx\)=\(\frac{1}{5}\log_{}{|sin(5x+2)|}+c\)
  
7) \(\int_{}^{}sec(ax+b)tan(ax+b)dx\)=\(\frac{1}{a}sec(ax+b)+c\)
                      

Substituting a = 3, and b = 8, we get,
  \(\int_{}^{}sec(3x+8)tan(3x+8)dx\)=\(\frac{1}{a}sec(3x+8)+c\)
  
  Where C is intregration constant

8) \(\int_{}^{}cosec(ax+b).cot(ax+b)dx\)=\(-\frac{1}{a}cosec(ax+b)+c\)
                      

Substituting a = 4, and b = 5, we get,
  \(\int_{}^{}-cosec(4x+5).cot(4x+5)dx\)=\(-\frac{1}{-4}cosec(4x+5)+c\)
         =\(\frac{1}{4}cosec(4x+5)+c\)
         
 9)  \(\int_{}^{}sec(ax+b)dx\)=\(\frac{1}{a}log_{}{|sec(ax+b)+tan(ax+b)|}+c\)
                      =\(\frac{1}{a}log_{}{|tan(\frac{\pi}{4}+\frac{ax+b}{2})|}+c\)
                      
Substituting a = 3, and b = 8, we get,
  \(\int_{}^{}sec(3x+8)dx\)=\(\frac{1}{3}log_{}{|sec(3x+8)+tan(3x+8)|}+c\)
                      =\(\frac{1}{3}log_{}{|tan(\frac{\pi}{4}+\frac{3x+8}{2})|}+c\)

10) \(\int_{}^{}cosec(ax+b)dx\)=\(-\frac{1}{a}log_{}{|cosec(ax+b)-cot(ax+b)|}+c\)
           =\(-\frac{1}{a}log_{}{|tan\frac{1}{2}(ax+b)|}+c\)              

Substituting a = 8, and b = 9, we get,
 \(\int_{}^{}cosec(8x+9)dx\)=\(-\frac{1}{8}log_{}{|cosec(8x+9)-cot(8x+9)|}+c\)
           =\(-\frac{1}{8}log_{}{|tan\frac{1}{2}(8x+9)|}+c\)