If ‘f’ and ‘g’ are functions of ‘x’, such that g'(x)=f(x) then the function ‘g’ is called an integral of ‘f’ with respect to ‘x’, and is written symbolically as:

f(x)dx = g(x) + c

where: f(x) is called the integrand and ‘c’ is called the constant of integration

Note: If d/dx f(x) = g(x) then d/dx {f(x) + c} = g(x)

Where ‘c’ is constant, because differentiation of a constant is zero.

Thus the general value g(x)dx is f(x)+c, where ‘c’ is the constant of integration.

Clearly integral will change if ‘c’ changes. Thus integral of a function is not unique, hence it is called indefinite integral.

### Standard Results:

These standard results for integral calculus are derived directly from the standard results of differential calculus

Differential Calculus | Integral Calculus |
---|---|

d/dx(x^{n+1}/
n+1) = x^{n} |
x^{n
}dx =(x^{n+1}/n+1) + C [n not
=1] |

d/dx log_{e}|x|
= 1/x |
1/x dx
= log_{e}|x| + c [n= -1] |

d/dx e^{x}
= e^{x} |
e^{x
}dx = e^{x} + c |

d/dx a^{x}
= a^{x} log_{e}a |
a^{x}
dx = a^{x} / log_{e}a + c
[a>0] |

d/dx Cosx = – Sinx | Sinx dx = – Cosx +c |

d/dx Sinx = Cosx | Cosx dx = Sinx + c |

d/dx Tanx =
Sec^{2}x |
Sec^{2}x
dx = Tanx + c |

d/dx Cotx =
– Cosec^{2} x |
Cosec^{2}x
dx = – Cotx + C |

d/dx Secx = Secx.Tanx | Secx.Tanx dx = Secx + c |

d/dx Cosecx = – Cosecx.Cotx | Cosec.Cotx dx = – Cosecx + c |

d/dx Sin^{-1}x
= 1/v(1-x^{2}) |
1/(1-x^{2})
dx = Sin^{-1} + c |

d/dx Tan^{-1}x
= 1/(1+x^{2}) |
1/(1+x^{2})
dx = Tan^{-1}x + c |

d/dx Sec^{-1}x
= 1/xv(x^{2} – 1) |
1/(x^{2}
– 1) dx = Sec^{-1} x + C |

d/dx Sin^{-1}x/a
= 1/v(a^{2} + x^{2}) |
dx/v(a^{2}
– x^{2}) = Sin^{-1}x/a + c |

d/dx (1/a)
Tan^{-1}x/a = 1/(x^{2}+a^{2}) |
dx/(x^{2}+a^{2})
= 1/a Tan^{-1}(x/a) +c |

d/dx (1/a
Sec^{-1}x/a) =1/xv(x^{2} – a^{2}) |
dx/xv(x^{2}-a^{2})
= 1/a Sec^{-1}x/a +c |

d/dx Coshx = Sinhx | Sinh dx = Coshx + c |

d/dx Sinhx = Coshx | Coshx dx = Sinhx + c |

d/dx Tanhx
= Sech^{2}x |
Sec^{2}x
dx = Tanhx + c |

d/dx Cothx
= – Cosech^{2}x |
Cosech^{2}x
dx = – Cothx +c |

d/dx Sechx = – Sechx.Tanhx | Sechx.Tanhx dx = – Sechx + c |

d/dx Cosechx = – Cosechx.Cothx | Cosechx.Cothx dx= -Cosechx+c |