Properties of Definite Integral





Properties of Definite Integral


Definite integral is part of integral or anti-derivative from which we get fixed answer rather than the range of answer or indefinite answers.


Some of the important formulas are shown below:-

1. \displaystyle{\int ^b_a f(x) dx = \int ^b_a f(t) dt \, \, \, (t \, \, is\, \, dummy \, \, variable)} \\ 2. \displaystyle{\int ^b_a f(x) dx = -\int ^a_b f(x) dx} \\ 3. \displaystyle{\int ^b_a f(x) dx = \int ^c_a f(x) dx + \int ^b_c f(x) dx, \, \, \, a < c < b} \\ 4. \displaystyle{\int ^a_0 f(x) dx = \int ^a_0 f(a- x)dx} \\ 5. \displaystyle{\int ^na_0 f(x) dx = n \int^n_0 f(x) dx} \\ 6. \displaystyle{\int ^2a_0 f(x) dx = 2\int ^n_0 f(x) dx} \\ 7. \displaystyle{\int ^n_a f(x) dx = 2\int ^n_a f(x) dx}

Note:

  1.  Even function: a function f(x) is called even function if f (-x) = f(x).
  2. A function f(x) is called odd function if f (-x) = -f(x).



Some standard relations

1. \displaystyle{\int ^\infty_a f(x)dx = \lim_{b \to \infty} \int ^b_a f(x) dx}

2. \displaystyle{\int ^b_a f(x) dx = \lim_{a \to \infty}\int ^b_a f(x) dx} \\

3. \displaystyle{\int ^\infty_\infty f(x) dx = \int ^a_\infty f(x) dx + \int ^\infty_a f(x) dx}

4. If \, f(x) \to \infty, \, \, as \, \, x \to z \\ \displaystyle{\int ^b_a f(x) dx = \lim_{n \to 0} \int ^b_{a + n} f(x) dx, h >0} \\

5. If \, f(x) \to \infty, \, \, as \, \, x \to b \\ \displaystyle{\int ^b_a f(x) dx = \lim_{h to 0} \int ^{b- h}_a f(x) dx, h > 0} \\

6. If  \, f(x) \to \infty, \, \, as \, \, x \to c, \, \, (a < c <b) \\ \displaystyle{\int ^b_a f(x) dx = \lim_{h \to 0} \int ^{c- h}_a f(x) dx + \lim_{h \to o} \int ^b_{c + h} f(x) dx} \\

7. If  \, f(x) \to \infty, \, when \, x \to a, \, x \to b \\ \displaystyle{\int ^b_a f(x) dx = \int ^c_a f(x) dx + \int ^b_c f(x) dx}, \, \, a < c < b \\

 8. \displaystyle{\int ^\infty_0 e^{-ax} \cos bx dx + \dfrac{a}{a^2 + b^2}} \\

9. \displaystyle{\int ^\infty_0 e^{-x}x^n dx = n !} \\

10. \displaystyle{\int ^\infty_0 \dfrac{\sin bx}{x} dx + \dfrac{\pi}{2}} (b > 0)

11. \displaystyle{\int ^\infty_0 e^{-x^2} dx + \dfrac{\sqrt{\pi}}{2}}



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