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(xn+1/ n+1) = xn | xn
dx =(xn+1/n+1) + C [n not
=1] |
| d/dx loge|x| = 1/x | 1/x dx
= loge|x| + c [n= -1] |
| d/dx ex = ex | ex
dx = ex + c |
| d/dx ax = ax logea | ax
dx = ax / logea + c
[a>0] |
| d/dx Cosx = – Sinx | Sinx dx
= – Cosx +c |
| d/dx Sinx = Cosx | Cosx dx
= Sinx + c |
| d/dx Tanx = Sec2x | Sec2x
dx = Tanx + c |
| d/dx Cotx = – Cosec2 x | Cosec2x
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-1x = 1/v(1-x2) | 1/(1-x2)
dx = Sin-1 + c |
| d/dx Tan-1x = 1/(1+x2) | 1/(1+x2)
dx = Tan-1x + c |
| d/dx Sec-1x = 1/xv(x2 – 1) | 1/(x2
– 1) dx = Sec-1 x + C |
| d/dx Sin-1x/a = 1/v(a2 + x2) | dx/v(a2
– x2) = Sin-1x/a + c |
| d/dx (1/a) Tan-1x/a = 1/(x2+a2) | dx/(x2+a2)
= 1/a Tan-1(x/a) +c |
| d/dx (1/a Sec-1x/a) =1/xv(x2 – a2) | dx/xv(x2-a2)
= 1/a Sec-1x/a +c |
| d/dx Coshx = Sinhx | Sinh dx
= Coshx + c |
| d/dx Sinhx = Coshx | Coshx
dx = Sinhx + c |
| d/dx Tanhx = Sech2x | Sec2x
dx = Tanhx + c |
| d/dx Cothx = – Cosech2x | Cosech2x
dx = – Cothx +c |
| d/dx Sechx = – Sechx.Tanhx | Sechx.Tanhx
dx = – Sechx + c |
| d/dx Cosechx = – Cosechx.Cothx | Cosechx.Cothx
dx= -Cosechx+c |