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Physics Topics can be both theoretical and experimental, with scientists using a range of tools and techniques to understand the phenomena they investigate.
What is the Symbol for Definite Integral?
Integration is the inverse process of differentiation. Suppose f(x) is a function of x and
\(\frac{d}{d x}\){f(x)} = F(x)
i.e., the derivative of f(x) with respect to x is F(x), and F(x) is also a function of x.
It may be said that the integral of Fx) with respect to x is f(x) and it can be expressed by the equation
∫F(x)dx = f(x) ……… (1)
F(x) is called the integrand. ‘∫’ and ‘dx’ are the symbols of integration.
Constant of integration: It is known,
\(\frac{d}{d x}\)(x5) = 5x4
and \(\frac{d}{d x}\)(x5 + c) = 5x4 [as c is a constant]
So, the derivatives of functions x5 and x5 + c are the same. Then the integration of 5x4 should be written in general as x5 + c, because here c is a constant and when c = 0, we get the function x5.
So, ∫5x4dx = 5 ᐧ \(\frac{x^{4+1}}{4+1}\) + c = x5 + c
This constant c is called the integration constant. As this constant is indefinite, it is called the indefinite integration constant.
Definite Integral: For definite integral, equation number (1) can be written as
\(\int_a^b\)F(x)dx = f(b) – f(a)
So if the value of x changes from a to b, then the value of f(x) changes by f(b) – f(a). This [f(b) – f(a)] is called the definite integral of F(x) within the limits a and b. Here b is called the upper limit and a is called the lower limit.
Definite integral can be described as area under the curve [Fig.]. \(\int_a^b\)f(x)dx is the area confined within the lines y = f(x), x axis, x = a and x = b i.e., this area may be written as area = \(\lim _{\Delta x_i \rightarrow 0} \sum_i f\left(x_i\right) \Delta x_i\) = \(\int_a^b f(x) d x\)
The symbol of integration i.e., ‘∫’ comes from the first letter of ‘summation’ and it is written as ‘long S’.
Basic properties of integration:
i) Integration of the product of a function and a constant:
If f(x) = ag(x) then,
∫f(x)dx = ∫ag(x)dx = a∫g(x)dx
ii) Integration of the sum or difference of two functions:
If f(x) = g(x) ± h(x) then,
∫f(x)dx = ∫{g(x) ± h(x)}dx
= ∫g(x)dx ± ∫h(x)dx
Example:
f(x) = 3x4 – 6x2 + 8x – 5
∴ ∫f(x) dx = ∫3x4dx – ∫6x2dx + ∫8xdx – ∫5dx
= 3 × \(\frac{x^{4+1}}{4+1}\) – 6 × \(\frac{x^2+1}{2+1}\) + 8 × \(\frac{x^{1+1}}{1+1}\) – 5 × \(\frac{x^{0+1}}{0+1}\) + c
= \(\frac{3}{5}\)x5 – 2x3 + 4x2 – 5x + c
iii) Interchange of upper limit and lower limit of a definite integral:
\(\int_a^b\)f(x)dx = –\(\int_b^a\)f(x)dx
Example:
iv) Insertion of any intermediate limit between the upper and lower limits:
\(\int_a^b\)f(x)dx = \(\int_a^c\)f(x)dx + \(\int_c^b\)f(xdx)
[c may be greater or less than both the upper and lower limits b and a]
Example:
Integrals of algebraic functions:
- ∫xndx = \(\frac{x^{n+1}}{n+1}\) + c (n ≠ -1)
- ∫axndx = a∫xndx = \(\frac{a x^{n+1}}{n+1}\) + c (n ≠ -1)
- ∫\(\frac{d x}{x}\) = ln|x| + c
- ∫amxdx = \(\frac{a^{m x}}{m \ln a}\) + c (a > 0, a ≠ 1)
Integrals of trigonometric functions:
- ∫sinxdx = -cosx + c
- ∫cosxdx = sinx + c
- ∫tanxdx = ln|secx| + c
- ∫cotxdx = ln|sinx| + c
- ∫secxdx = ln|secx + tanx| + c = ln|tan\(\left(\frac{\pi}{4}+\frac{x}{2}\right)\)| + c
- ∫cosecxdx = ln|cosecx – cotx| + c = ln|tan\(\frac{x}{2}\)| + c
- ∫sinmxdx = –\(\frac{\cos m x}{m}\) + c
- ∫cosmxdx = \(\frac{\sin m x}{m}\) + c
- ∫sec2xdx = tanx + c
- ∫cosec2xdx = -cotx + c
- ∫secxtanxdx = secx + c
- ∫cosecxcotxdx = -cosecx + c
Integrals of logarithmic and exponential functions:
- ∫ln axdx = x(lnax – 1) + c
- ∫exdx = ex + c
- ∫emxdx = \(\frac{e^{m x}}{m}\) + c