Twelve Standard >> List of some frequently used formula in integration | Part-1

List of some frequently used formula in integration

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

Substituting a = 2, b = 5, and n = 7, we get,
     \(\int_{}^{} (2x+5)^{7}dx\)=\(\frac{(2x+5)^{7+1}}{2(7+1)}+c\)
                         =\(\frac{(2x+5)^{8}}{16}+c\)
                        

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

      Substituting a = 5 and b =8, we get,
     \(\int_{}^{}\frac{dx}{5x-8} \)=\(\frac{1}{5}\log_{}{|5x-8|}+c\)

3) \(\int_{}^{}e^{ax+b}dx\)=\(\frac{e^{ax+b}}{a}+c\)

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

4) \(\int_{}^{}a^{bx+k}dx\)=\(\frac{a^{bx+k}}{b\log_{}{a}}+c\)

    Substituting a = 5, b = 5, and k = 6, we get,
     \(\int_{}^{}5^{5x+6}dx\)=\(\frac{5^{5x+6}}{5\log_{}{5}}+c\)